\section{Introduction}
Geometric measures associated with convex bodies have been the core part of convex geometric analysis in the past few decades. In the classical Brunn-Minkowski theory of convex bodies, \emph{quermassintegrals} (such as volume, surface area, mean width, and much more in higher dimensions) are the central geometric invariants and are used to describe the shape of convex bodies via isoperimetric (or reverse isoperimetric) inequalities. \emph{Area measures} introduced by Aleksandrov, Fenchel-Jessen, and \emph{curvature measures} introduced by Federer can be viewed as the ``derivative'' of quermassintegrals when viewed as functionals on the set of convex bodies. Invariably, these geometric measures carry some curvature terms which make it possible for them to encode shape information of convex bodies. At the same time, unlike curvatures (in the sense of differential geometry), these geometric measures are defined even with minimal regularity assumptions. The study of these geometric measures are often intertwined with PDE (Monge-Amp\`{e}re equations in particular), Gauss curvature flows, and inevitably isoperimetric inequalities. (After all, a half of calculus is focused on using derivative to study properties of the original function.)

A major alternative to the classical Brunn-Minkowski theory in modern convex geometry is the dual Brunn-Minkowski theory. The dual Brunn-Minkowski theory, introduced by Lutwak in 1975, is a theory that is in a sense dual to the classical Brunn-Minkowski theory. A good discussion of the dual theory can be found in Section 9.3 of Schneider's classical volume \cite{MR3155183}. Quoting from Gardner-Hug-Weil \cite{MR3120744}:``The dual Brunn-Minkowski theory can count among its successes the solution of the Busemann-Petty problem in \cite{MR1298719}, \cite{MR1689343}, \cite{MR963487}, and \cite{MR1689339}. It also has connections and applications to integral geometry, Minkowski geometry, the local theory of Banach spaces, and stereology; see \cite{MR2251886} and the references given there.'' The dual theory studies interior properties of convex bodies while the classical theory is most effective in dealing with boundary information of convex bodies.

In the seminal work \cite{MR3573332}, Huang-Lutwak-Yang-Zhang (Huang-LYZ), for the first time, revealed the fundamental geometric measures, duals of Federer's curvature measures, called \emph{dual curvature measures}, in the dual Brunn-Minkowski theory. These measures were obtained through ``differentiating'' \emph{dual quermassintegrals} which are central in the dual theory. They have led to many natural open problems and quickly attracted much attention. Details on this will be provided below.

It is well-known that the set of convex bodies can be embedded in the set of upper semi-continuous log-concave functions via their characteristic functions. It is the aim of this work to introduce the functional version of dual curvature measures through the machinery of the theory of functions of bounded anisotropic weighted variation and to study their characterization problem (generally known as \emph{Minkowski problems}). It is worth pointing out that, by picking proper weight functions, functional versions of many other geometric measures can be introduced in this fashion. However, this will not be explored in this work.

In the past few decades (even more so in the last decade), interest in log-concave functions have grown considerably, much of it motivated by their counterparts in the theory of convex bodies. Perhaps the first such breakthrough and by now a well-known result is the Pr\'{e}kopa-Leindler inequality: For any nonnegative integrable functions $f, g$ on ${\mathbb R^n}$ and their sup-convolution $(1-\lambda)\cdot f\oplus \lambda\cdot g$ given by
\begin{equation}
	(1-\lambda)\cdot f\oplus \lambda\cdot g (z)= \sup_{(1-\lambda)x+\lambda y=z} f(x)^{1-\lambda}g(y)^\lambda,
\end{equation}
where $0<\lambda<1$, one has the following inequality regarding their $\L^1$ norms,
\begin{equation}
	\int_{\mathbb R^n} (1-\lambda)\cdot f\oplus \lambda\cdot g (z)dz\geq \left(\int_{{\mathbb R^n}} f(x)dx\right)^{1-\lambda}\left(\int_{\mathbb R^n} g(y)dy\right)^\lambda.
\end{equation}
The Pr\'{e}kopa-Leinder inequality is the functional (and equivalent) version of the celebrated Brunn-Minkowski inequality,
\begin{equation}
	V((1-\lambda)X+\lambda Y)\geq V(X)^{1-\lambda}V(Y)^\lambda,
\end{equation}
which holds for any bounded measurable sets $X, Y\subset {\mathbb R^n}$ such that $(1-\lambda)X+\lambda Y$ is measurable.
 See the survey \cite{MR1898210} by Gardner. It is important to note that convexity is required neither in the Brunn-Minkowski inequality, nor in the Pr\'{e}kopa-Leindler inequality, although it does manifest itself in the equality conditions. Notice here that the Minkowski combination $(1-\lambda)X+\lambda Y$ corresponds the sup-convolution between functions and the volume of a measurable set corresponds the integral of a non-negative function. That this is natural can be seen by replacing $f$ and $g$ by characteristic functions of subsets of ${\mathbb R^n}$.

 In the same spirit, many other geometric invariants and operations have found their counterparts for log-concave functions (or, equivalently, convex functions). We provide a quick overview of some of these remarkable results. In the seminal work \cite{MR2480615}, Artstein-Avidan and V. Milman demonstrated how the Legendre-Fenchel transform can be viewed as the functional version of taking the polar body of a convex body. Compare this to another remarkable paper \cite{MR2438994} by B\"{o}r\"{o}czky-Schneider. Prior to this, the functional version of the Blaschke-Santal\'o inequality was discovered by K. Ball in his Ph.D. thesis and Artstein-Avidan, Klartag, and V. Milman \cite{MR2220210}. Steiner formula and quermassintegrals for quasi-concave functions were studied by Bobkov-Colesanti-Fragal\`{a} \cite{MR3147446}. Extensions of affine surface area and affine isoperimetric inequalities can be found in Artstein-Avidan-Klartag-Sch\"{utt}-Werner \cite{MR2899992}, Caglar-Werner \cite{MR3335279}, Caglar-Ye \cite{MR3551664}, and Lin \cite{MR3695895}. Much more recently, Colesanti, Ludwig, and Mussnig embarked on a journey to characterize valuations on the set of convex functions \cite{MR4453228,https://doi.org/10.48550/arxiv.2109.09434,https://doi.org/10.48550/arxiv.2009.03702,MR4124129,https://doi.org/10.48550/arxiv.2201.11565} (compare them to Hadwiger-type theorems on convex bodies \cite{MR2680490,MR2772547,MR1991649,MR2966660,MR3194492,MR2668553}).

In the dual Brunn-Minkowski theory, the central geometric invariants are known as \emph{dual quermassintegrals}. Let $q=1, \cdots, n$ and $K$ be a convex body that contains the origin in its interior. As Lutwak \cite{MR963487} showed, up to a constant multiple, the $q$-th dual quermassintegral $\widetilde{V}_q(K)$ can be defined as the average of lower-dimensional sectional areas of $K$ with $q$-dimensional subspaces:
\begin{equation}
	\widetilde{V}_q(K)=c\int_{G(n,q)}\mathcal{H}^{q}(K\cap \xi)d\xi,
\end{equation}
where $G(n,q)$ is the Grassmannian manifold containing all $q$-dimensional subspaces of ${\mathbb R^n}$ and the integration is with respect to the Haar measure. Dual quermassintegrals have integral representations (see \eqref{eq 91}) which warrant the immediate extension to $q\in \mathbb{R}$. Note that with the exception of the special case $q=n$---when the dual quermassintegral is simply volume, the $q$-th dual quermassintegral is generally \emph{not} invariant under translation of $K$.

A major question answered in the landmark work \cite{MR3573332} (and subsequently \cite{MR3783409}) by Huang-LYZ was the differentiability of $\widetilde{V}_q$. In particular, it was shown that if $K$ is a convex body that contains the origin in its interior, then
\begin{equation}
\label{eq 92}
	\lim_{t\rightarrow 0}\frac{\widetilde{V}_q(K+ t\cdot L)-\widetilde{V}_q(K)}{t}= \int_{ {S^{n-1}}} h_L(v) \frac{1}{h_K(v)}d\widetilde{C}_q(K,v).
\end{equation}
Here the geometric measure $\widetilde{C}_q(K,\cdot)$ is known as the $q$-th \emph{dual curvature measure} of $K$. In fact, there is naturally an $\L_p$ version of \eqref{eq 92} that leads to the $(p,q)$-dual curvature measure introduced in \cite{MR3783409} and the measure $\frac{1}{h_K(v)}d\widetilde{C}_q(K,v)$ is nothing but the $(1,q)$-dual curvature measure. The ``1'' here stands for the fact that the sum $K+t\cdot L$ is the classical Minkowski sum, or, the $l^1$ sum of support functions of $K$ and $L$.

\mycomment{
 Minkowski problems on dual curvature measures and $(p,q)$-dual curvature measures are generally known as dual Minkowski problems. In a short period since \cite{MR3573332}, there have been many influential works on this topic  and have already led to many interesting conjectures regarding isoperimetric inequalities as well as the discovery of novel curvature flows; see, for example, B\"{o}r\"{o}czky-Henk-Pollehn \cite{MR3825606}, Chen-Chen-Li \cite{MR4259871}, Chen-Huang-Zhao \cite{MR3953117}, Chen-Li \cite{MR3818073}, Gardner-Hug-Weil-Xing-Ye \cite{MR3882970}, Henk-Pollehn \cite{MR3725875}, Li-Sheng-Wang \cite{MR4055992}, Liu-Lu \cite{MR4127893}, Zhao \cite{MR3880233}. It is important to note that the list is by no means exhaustive. }

Let $q>0$. The $(q-n)$-th moment of a density function $f$ is defined as
\begin{equation}
	\widetilde{V}_q(f) = \int_{{\mathbb R^n} }|x|^{q-n}f(x)dx,
\end{equation}
if it exists. The moment $\widetilde{V}_q$ is a natural extension of dual quermassintegrals to the set of log-concave functions (which in turn justifies this notation). Indeed, if $f=1_K$ where $K$ is some convex body that contains the origin in its interior, then, by integration via polar coordinates, one immediate has
\begin{equation}
	\widetilde{V}_q(1_K) = \widetilde{V}_q(K).
\end{equation}
Motivated by the work \cite{MR3573332} and the correspondence between Minkowski combination and sup-convolution, it is natural to ask whether
\begin{equation}
\label{eq 93}
	\lim_{t\rightarrow 0} \frac{\widetilde{V}_q(f\oplus t\cdot g)-\widetilde{V}_q(f)}{t}
\end{equation}
exists, for log-concave functions $f,g$ and if it does, what the limit is. We remark that with the exception of $q=n$, for generic $q>0$, the moments considered in \eqref{eq 93} is \emph{not} invariant under the transformation $f(x)\rightarrow f(x+x_0)$. Therefore, the relative position of the origin is crucial in the study of \eqref{eq 93}. In fact, since \eqref{eq 92} only holds when $K$ contains the origin in its interior, some condition on $f$ that mimics this constraint is expected.

When $q=n$, the functional $\widetilde{V}_q$ is nothing but the $\L^1$ norm of a log-concave function. In this case, the limit \eqref{eq 93} was studied by Colesanti-Fragal\`{a} \cite{MR3077887} under various regularity assumptions on $f$ and $g$. In particular, they discovered that the limit \eqref{eq 93} consists of two part---one concerning the behavior of $f$ inside its support, the other concerning the values of $f$ on the boundary of its support as well as the shape of the support set. Around the same time, Cordero-Erausquin and Klartag \cite{MR3341966} studied the limit with the additional assumption that $f$ is \emph{essentially continuous} and explored the connection with complex analysis \cite{MR3137248} and optimal transport.
Recently, Rotem \cite{https://doi.org/10.48550/arxiv.2206.13146} showed that the result of Colesanti-Fragal\`{a} remains valid without any of the various additional regularity requirements, by employing tools from the theory of functions of bounded anisotropic  variation. The first main result of this paper is to show that by considering functions of bounded anisotropic \emph{weighted} variation, one can compute the limit in \eqref{eq 93} for any $q>0$. It is important to emphasize that unlike the special $q=n$ case, for generic $q>0$, the $q$-th moment of a function is not translation-invariant. In particular, our approach is motivated by the dual approach (differentiating radial functions) to variational formula developed in \cite{MR3573332}.


It is also worth pointing out that the extension of functions of bounded variation in ${\mathbb R^n}$ with respect to Lebesgue measure to functions of bounded variation in ${\mathbb R^n}$ with an arbitrary measure (even those absolutely continuous with respect to Lebesgue measure) is not entirely trivial. This has been previously done in, for example, \cite{MR1864805} and \cite{MR2005202} via different approaches (which led to non-equivalent definitions of weighted total variation).

Let $\Lc({\mathbb R^n})$ be the set of all upper semi-continuous log-concave functions $f:{\mathbb R^n} \rightarrow [0,\infty)$. The limit \eqref{eq 93} leads to two Borel measures---one on ${\mathbb R^n}$ and one on $ {S^{n-1}}$.

 \begin{definition}
 \label{def intro 1}
	Let $f=e^{-\phi}\in \Lc({\mathbb R^n})$ with nonzero finite $\L^1$ norm. The Euclidean $q$-th dual curvature measure of $f$,  denoted by $\widetilde{C}^{e}_{q}(f;\cdot)$, is a Borel measure on ${\mathbb R^n}$ given by
	\begin{equation}
		\widetilde{C}^{e}_{q}(f;\mathcal{B}) = \int_{\nabla\phi(x)\in \mathcal{B}}|x|^{q-n}f(x)\,dx,
	\end{equation}
	for each Borel set $\mathcal{B}\subset {\mathbb R^n}$.
\end{definition}

\begin{definition}
\label{def intro 2}
	Let $f=e^{-\phi}\in \Lc({\mathbb R^n})$ with nonzero finite $\L^1$ norm. The spherical $q$-th dual curvature measure of $f$, denoted by $\widetilde{C}^{s}_{q}(f;\cdot)$, is a Borel measure on $ {S^{n-1}}$ given by
	\begin{equation}
		\widetilde{C}^{s}_{q}(f;\eta) = \int_{\nu_{K_f}(x)\in \eta}|x|^{q-n}f(x)\,d\mathcal{H}^{n-1},
	\end{equation}
	for each Borel set $\eta\subset  {S^{n-1}}$, where $K_f$ is the support of $f$.
\end{definition}
These two measures generated through differentiating the $q$-th moment of a log-concave function $f$ with respect to sup-convolution are associated with the absolutely continuous and singular part of the distributional derivative of $f$, respectively. It is worth noting that in the case of characteristic function of convex body, the measure $\widetilde{C}^{s}_{q}(f;\cdot)$ recovers the $(1,q)$-dual curvature measure for convex bodies appearing in \eqref{eq 92}.

The first of our main theorems shows that the limit in \eqref{eq 93} does exist under minor assumptions on $f$ near the origin.
\begin{theorem}
\label{theorem intro 1}
 		Let $f=e^{-\phi}\in \Lc({\mathbb R^n})$ with non-zero finite $\L^1$ norm and $q>0$. Assume $f$ achieves its maximum at the origin and
 		\begin{equation}
 		\label{eq 400a}
 			\limsup_{x\rightarrow o} \frac{|f(x)-f(o)|}{|x|^{\alpha+1}}<\infty,
 		\end{equation}
 		for some $0<\alpha<1$.
 		
 		 Let $g=e^{-\psi}\in \Lc({\mathbb R^n})$ be compactly supported. Then,
 		\begin{equation}
 		\label{eq 95}
	\lim_{t\rightarrow 0^+} \frac{\widetilde{V}_q(f\oplus t\cdot g)-\widetilde{V}_q(f)}{t} = \int_{{\mathbb R^n}}\psi^*(y)d\widetilde{C}^{e}_{q}(f;y)+\int_{ {S^{n-1}}} h_{K_g}(v)d\widetilde{C}^{s}_{q}(f;v).
 		\end{equation}
 		where $h_{K_g}$ is the support function of the support set $K_g$ of $g$, and $\psi^*$ is the Legendre-Fenchel conjugate of $\psi$.
 \end{theorem}
  We emphasize again that it is expected that we need some condition on $f$ that mimics the idea that ``$f$ contains the origin in its interior''. The assumption that $f$ achieves its maximum at the origin ensures that almost all of $f$'s nonempty level sets contain the origin in the interior. Equation \eqref{eq 400a} guarantees that these level sets contain the origin in the interior in some \emph{uniform} way. We remark that if $f$ is $C^{1,\alpha}$ in a neighborhood of the origin, then \eqref{eq 400a} is satisfied.


 When $f$ and $g$ are characteristic functions of convex bodies that contain the origin in their interiors, the first term in \eqref{eq 95} vanishes and the second term becomes the right side of \eqref{eq 92}.

One important aspect of the study of geometric measures arising in this fashion is through their \emph{Minkowski problems}. In differential geometry, they are known as various prescribed curvature problems. This line of research that asks when a given measure $\mu$ can be realized as a certain geometric measure of a to-be-solved convex body (without any unnecessary regularity assumptions) goes all the way back to the classical Minkowski problem that inspired the study of nonlinear elliptic PDE through the last century; see, for example, Minkowski \cite{MR1511220}, Aleksandrov \cite{MR0001597}, Cheng-Yau \cite{MR0423267}, Pogorelov\cite{MR0478079}, and  the works of Caffarelli on the regularity theory of Monge-Amp\`{e}re equations \cite{MR1005611, MR1038359,MR1038360}, among many other influential works. In many ways, the study of Minkowski problems goes hand-in-hand with the study of sharp isoperimetric inequalities; see \cite{MR1849187}.


In the last 2-3 decades, there are two major families of Minkowski problems. One is the $\L_p$ Minkowski problem that belongs to the $\L_p$ Brunn-Minkowski theory whose success can be credited to the landmark work by Lutwak \cite{MR1231704, MR1378681} where the fundamental $\L_p$ surface area measure was discovered. The $\L_p$ Minkowski problem includes the logarithmic Minkowski problem and the centro-affine Minkowski problem and have been studied through a variety of methods; see, for example, Hug-Lutwak-Yang-Zhang (Hug-LYZ) \cite{MR2132298}, Chou-Wang \cite{MR2254308}, B\"{o}r\"{o}czky-LYZ \cite{MR3037788}, and most recently Guang-Li-Wang \cite{https://doi.org/10.48550/arxiv.2203.05099}. A vast library of works on this topic can be found by looking for those citing the above mentioned works. It is worth pointing out that there is much unknown regarding the $\L_p$ Brunn-Minkowski theory, especially for $p<1$. In fact, the log Brunn-Minkowski conjecture, arguably the most beautiful and powerful (yet plausible) conjecture in convex geometry in the last decade, is the isoperimetric inequality associated with the log Minkowski problem. See, for example, \cite{MR2964630, MR4438690, MR4088419, milman2021centroaffine, vanhandel2022local}.

The other major family of Minkowski problems is the \emph{dual Minkowski problem} following the landmark work \cite{MR3573332}. In a short period since \cite{MR3573332}, there have been many influential works on this topic  and have already led to many interesting conjectures regarding isoperimetric inequalities as well as the discovery of novel curvature flows; see, for example, B\"{o}r\"{o}czky-Henk-Pollehn \cite{MR3825606}, Chen-Chen-Li \cite{MR4259871}, Chen-Huang-Zhao \cite{MR3953117}, Chen-Li \cite{MR3818073}, Gardner-Hug-Weil-Xing-Ye \cite{MR3882970}, Henk-Pollehn \cite{MR3725875}, Li-Sheng-Wang \cite{MR4055992}, Liu-Lu \cite{MR4127893}, Zhao \cite{MR3880233}. It is important to note that the list is by no means exhaustive.

In this paper, we study the Minkowski problem for $\widetilde{C}^{e}_{q}$.

\textbf{The functional dual Minkowski problem.} Let $q>0$ and $\mu$ be a non-zero finite Borel measure on ${\mathbb R^n}$. Find the necessary and sufficient conditions on $\mu$ so that there exists $f\in \Lc({\mathbb R^n})$ with nonzero finite $\L^1$ norm such that
\begin{equation}
\label{eq 96}
	\mu = \widetilde{C}^{e}_{q}(f;\cdot).
\end{equation}

Under sufficient regularity assumption; that is, the measure $\mu$ has a $C^\infty$ density (say, $g\geq 0$) and $f\in C^{\infty}$, equation \eqref{eq 96} is equivalent to the following Monge-Amp\`{e}re type equation in ${\mathbb R^n}$
\begin{equation}
\label{eq 99}
	g(\nabla \phi(x))\det (\nabla^2 \phi(x)) = |x|^{q-n}e^{-\phi(x)},
\end{equation}
where $f=e^{-\phi}$.

It is important to note that the measure $\widetilde{C}^{e}_{q}(f;\cdot)$ might not be absolutely continuous. Thus, the Minkowski problem \eqref{eq 96} does not always reduce to \eqref{eq 99} in the general setting.

The case $q=n$ is the Minkowski problem for moment measures within the class of essentially continuous functions was fully solved in Cordero-Erausquin and Klartag \cite{MR3341966}. The highly nontrivial $\L_p$ extension of Cordero-Erausquin and Klartag's result can be found in the recent papers by Fang-Xing-Ye \cite{MR4375791} for $p>1$ and Rotem \cite{https://doi.org/10.48550/arxiv.2006.16933} for $0<p<1$.

 As have pointed out earlier, a key difference between the case $q=n$ and $q\neq n$ is that in the latter case, \eqref{eq 96} is not invariant under translation of $f$ (with respect to its domain). We point out that translation-invariance played a central role in \cite{MR3341966}.

In the current work, we solve \eqref{eq 96} in the origin-symmetric case.
\begin{theorem}
\label{theorem intro 2}
	Let $q>0$ and $\mu$ be a non-zero even finite Borel measure on ${\mathbb R^n}$. Suppose $\mu$ is not entirely concentrated in any proper subspaces and $\int_{\mathbb R^n} |x|d\mu(x)<\infty$. There exists an even $f_0\in \Lc({\mathbb R^n})$ with nonzero finite $\L^1$ norm such that
	\begin{equation}
		\mu = \widetilde{C}^{e}_{q}(f_0;\cdot).
	\end{equation}
\end{theorem}

As can be expected, the functional dual Minkowski problem \eqref{eq 96} is heavily intertwined with its counterpart in the setting of convex bodies. In particular, estimates regarding dual quermassintegrals are critically needed. As part of the process to get the required estimates, we require a Blaschke-Santal\'{o} type inequality for the functional $\widetilde{V}_q$. See Lemma \ref{lemma 4}. It is of great interest to see if there is a sharp (more refined) version.

We remark here that,  the assumption that
\begin{equation}
	\int_{\mathbb R^n} |x|d\mu(x)<\infty
\end{equation}
is necessary here. See Theorem \ref{thm necessity}.

The rest of this paper is organized in the following way. In Section 2, we recall some notations and basics. In Section 3, we gather some basics in the theory of functions of bounded variation. Section 4 is devoted to prove Theorem \ref{theorem intro 1} whereas Section 5 is devoted to prove Theorem \ref{theorem intro 2}.

\section{Preliminaries}

This section is divided into two parts. The first part contains some notations and basics in the theory of convex bodies, whereas the second part deals with those for convex functions as well as log-concave functions.

\subsection{Convex bodies}
\label{section preliminary convex body}
The standard reference is the comprehensive book \cite{MR3155183} by Schneider.

A convex body in ${\mathbb R^n}$ is a compact convex set with nonempty interior. The boundary of $K$ is written as $\partial K$. We use $\mathcal{K}^n$ for the set of all convex bodies in ${\mathbb R^n}$. The subclass of  convex bodies that contain the origin in their interiors in ${\mathbb R^n}$ is denoted by $\mathcal K_o^n$.

We will use $B(x,r)$ to denote that ball in ${\mathbb R^n}$ centered at $x$ with radius $r$. Occasionally, we write $B(r)=B(o,r)$ and $B=B(o,1)$ for simplicity.

The support function $h_K$ of $K$ is defined by
\begin{equation}
\label{eq local 0004}
	h_K(y) = \max\{x\cdot y : x\in K\}, \quad y\in{\mathbb R^n}.
\end{equation}
The support function $h_K$ is a continuous function homogeneous of degree 1. Suppose $K$ contains the origin in its interior. The radial function $\rho_K$ is defined by
\[ \rho_K(x) = \max\{\lambda : \lambda x \in K\}, \quad x\in {\mathbb R^n}\setminus \{0\}. \]
The radial function $\rho_K$ is a continuous function homogeneous of degree $-1$. It is not hard to see that $\rho_K(u)u \in
\partial K$ for all $u\in S^{n-1}$ and the reciprocal radial function is a (potentially asymmetric) norm. To be more specific, we write
\begin{equation}
\label{eq 210}
	\|x\|_K = \frac{1}{\rho_K(x)} = h_{K^*}(x), \quad \text{for each } x\in {\mathbb R^n}.
\end{equation}
Here, the convex body $K^*$ is known as the polar body of $K$ and is defined by
\begin{equation}
	K^*= \{y\in {\mathbb R^n}: \langle x,y\rangle\leq 1, \forall x\in K\}.
\end{equation}
By the definition of the polar body, it is simple to see that the Banach space $({\mathbb R^n}, \|\cdot\|_{K})$ and $({\mathbb R^n}, \|\cdot\|_{K^*})$ are dual to each other and we have the following generalized Cauchy-Schwarz inequality
\begin{equation}
	\langle x, y\rangle \leq \|x\|_K\|y\|_{K^*}.
\end{equation}

For each $x\in \partial K$, we will write $\nu_K(x)$ for the outer unit normal of $K$ at $x$. Note that by convexity, the map $\nu_K$ is defined almost everywhere on $\partial K$ with respect to $\mathcal{H}^{n-1}$. Similarly, we will write $\nu_K^{-1}(v)$ to denote the inverse map of $\nu_K$ and by convexity, $\nu_K^{-1}$ is almost everywhere defined on $ {S^{n-1}}$.

The fundamental geometric functionals in the dual Brunn-Minkowski theory are dual quermassintegrals. For $q\neq 0$, the $q$-th dual quermassintegral of $K$, denoted by $\widetilde{V}_q(K)$ is defined as
\begin{equation}
\label{eq 91}
	\widetilde{V}_q(K) = \frac{1}{q}\int_ {S^{n-1}} \rho_K^q(u)du.
\end{equation}
When $q=1,\cdots, n$, quermassintegrals have the strongest geometric significance. They are propotional to the mean of the $q$-dimensional volume of intersections of $K$ with $q$-dimensional subspaces in ${\mathbb R^n}$. We will use
\begin{equation}
	\overline{V}_{q}(K) = \left(\frac{1}{q}\int_ {S^{n-1}} \rho_K^q(u)du\right)^{\frac{1}{q}},
\end{equation}
for the normalized version of dual quermassintegrals.

In \cite{MR3573332}, it was established that variation of the dual quermassintegral with respect to the logarithmic Minkowski sum leads to the so-called dual curvature measure:
\begin{equation}
	\widetilde{C}_q(K,\eta) = \int_{\nu_K(x)\in \eta} x\cdot \nu_K(x) |x|^{q-n}d\mathcal{H}^{n-1}(x), \text{ for each Borel } \eta\subset  {S^{n-1}}.
\end{equation}
In particular, this implies that for each $p\in \mathbb{R}$, we have
\begin{equation}
\label{eq 71}
	\lim_{t\rightarrow 0}\frac{\widetilde{V}_q(K+_pt\cdot L)-\widetilde{V}_q(K)}{t} = \int_ {S^{n-1}} h_L(v)h_K^{-p}(v)d\widetilde{C}_{q}(K,v):=\int_{ {S^{n-1}}}h_L(v)d\widetilde{C}_{p,q}(K,v),
\end{equation}
where the Borel measure $\widetilde{C}_{p,q}(K,\cdot)$ is known as the $(p,q)$-dual curvature measure of $K$.
Here, the combination $K+_pt\cdot L$ is known as the $\L_p$ Minkowski combination between convex bodies. The variational formula, as well as the definition of the $\L_p$ combination can be found in \cite{MR3783409}.

\subsection{Convex functions and log-concave functions}Let $\cvx({\mathbb R^n})$ be the set of all lower semi-continuous, convex functions $\phi:{\mathbb R^n} \rightarrow (-\infty, \infty]$ and  $\Lc({\mathbb R^n})$ be the set of all upper semi-continuous log-concave functions $f$ that take the form $f=e^{-\phi}$ for some $\phi\in \cvx({\mathbb R^n})$.

For any function $\phi:{\mathbb R^n} \rightarrow [-\infty, \infty]$, the Legendre-Fenchel conjugate of $\phi$, denoted by $\phi^*$ is defined as
\begin{equation}
	\phi^*(y) = \sup_{x\in {\mathbb R^n}} \{\langle x, y\rangle - \phi(x)\}.
\end{equation}
Note that from the definition, it is simple to see that $\phi^*\in \cvx({\mathbb R^n})$, as long as $\phi\not\equiv +\infty$. It is also straightforward from the definition that Legendre-Fenchel transform reverses order; in another word, if $\phi_1\leq \phi_2$, then $\phi_1^*\geq \phi_2^*$.

When restricting to $\cvx({\mathbb R^n})$, Fenchel--Moreau theorem states that the Legendre-Fenchel transform is an involution:
\begin{equation}
	\phi^{**}=\phi, \text{ for each }\phi\in \cvx({\mathbb R^n}).
\end{equation}

In a remarkable paper \cite{MR2480615}, Arstein-Avidan and V. Milman showed that any order-reversing involution on $\cvx({\mathbb R^n})$ is essentially the Legendre-Fenchel transform.

It is also useful to consider the double Legendre-Fenchel transform for $\phi\notin \cvx({\mathbb R^n})$. In that case, the Legendre-Fenchel transform is no longer an involution. However, from the definition, one may show that
\begin{equation}
	\phi^{**}\leq \phi,
\end{equation}
and if $\phi\geq 0$, then $\phi^{**}\geq 0$.

We shall require the following trivial facts.
\begin{prop}
\label{prop 12}
	Let $f=e^{-\phi}\in \Lc({\mathbb R^n})$ and $q>0$. If
\begin{equation}
\label{eq 500}
	\liminf_{|x|\rightarrow \infty} \frac{\phi(x)}{|x|}>0,
\end{equation}	
then
\begin{equation}
	\widetilde{V}_q(f)=\int_{\mathbb R^n} f(x)|x|^{q-n}dx<\infty.
\end{equation}
\end{prop}
\begin{proof}
By \eqref{eq 500}, there exists $C>0$ and $r_0>0$ such that
\begin{equation}
	\phi(x)>C|x|, \text{ for all } x\in B(r_0)^c.
\end{equation}
Therefore, we have
\begin{equation}
	\int_{B(r_0)^c} f(x)|x|^{q-n}dx\leq \int_{B(r_0)^c} e^{-C|x|}|x|^{q-n}dx<\infty.
\end{equation}
Since $f$ is upper semi-continuous, it is locally bounded from above. Therefore,
\begin{equation}
		\int_{B(r_0)} f(x)|x|^{q-n}dx\leq C_1\int_{B(r_0)}|x|^{q-n}dx<\infty.
	\end{equation}
\end{proof}

It is well-known that \eqref{eq 500} holds if and only if either of the following two statements holds:
\begin{enumerate}
	\item $f\in \L^1$;
	\item $\lim_{|x|\rightarrow \infty}\phi(x)=\infty$.
\end{enumerate}
See, for example, \cite{MR3341966}.
\begin{prop}
\label{prop 51}
	Let $q>0$. If $\phi:{\mathbb R^n}\rightarrow [0,\infty]$ is finite in a neighborhood of the origin, then
	\begin{equation}
		\widetilde{V}_q(e^{-\phi^*})=\int_{\mathbb R^n} |y|^{q-n} e^{-\phi^*(y)}dy<\infty.
	\end{equation}
\end{prop}
\begin{proof}
	 We may assume without loss of generality that $\phi \in \cvx({\mathbb R^n})$, since $\phi\geq \phi^{**}$ and $(\phi^{**})^* = \phi^*$.
	
	 Suppose $r_0>0$ is such that $\phi$ is finite for all $|x|<r_0$. Since $\phi$ is convex on $\{x: |x|<r_0\}$, we have that $\phi$ is continuous there and there exists $M>0$ such that
	 \begin{equation}
	 	|\phi(x)|<M,
	 \end{equation}
	 for all $|x|\leq \frac{r_0}{2}$. For each $y\in {\mathbb R^n}$, set $x_y = \frac{y}{|y|}\cdot \frac{r_0}{2}$. Then, by definition of Legendre-Fenchel transform, we have
	 \begin{equation}
	 	\phi^*(y)\geq x_y \cdot y-\phi(x_y)\geq \frac{r_0}{2}|y|-M.
	 \end{equation}
	 Hence,
	 \begin{equation}
	 	\int_{\mathbb R^n} |y|^{q-n} e^{-\phi^*(y)}dy\leq \int_{{\mathbb R^n}}|y|^{q-n}e^{-\frac{r_0}{2}|y|}e^Mdy<\infty.
	 \end{equation}
\end{proof}
	
\section{Functions of anisotropic weighted total variation}
\label{sec BV}


Let $\Omega\subset {\mathbb R^n}$ be an open set. We say a function $f\in \L^1_{\loc}(\Omega)$ is locally of bounded variation (\emph{i.e.}, $f\in BV_{\loc}(\Omega)$) if for each open $V\Subset \Omega$, we have
\begin{equation}
\label{eq 101}
	TV(f; V) = \sup\left\{\int_{V} f\Div T\, dx: T\in C_c^1(V, {\mathbb R^n}), \|T(x)\|\leq 1, \forall x\right\}<\infty.
\end{equation}
Intuitively speaking, functions of locally bounded variation are those whose distribution derivatives are Radon measures. Indeed, the Structure Theorem for $BV_{\loc}$ functions states (see Theorem 5.1 in \cite{MR3409135}) that if $f\in BV_{\loc}(\Omega)$, then there exists a Radon measure $\|Df\|$ on $\Omega$ and a $\|Df\|$-measurable maps $\sigma:\Omega\rightarrow {\mathbb R^n}$ such that $\|\sigma\|=1$, $\|Df\|$-almost everywhere with
\begin{equation}
\label{eq 104}
	\int_{\Omega} f\Div T\, dx = -\int_\Omega \langle T, \sigma\rangle d\|Df\|,
\end{equation}
for all $T\in C_c^1(\Omega, {\mathbb R^n})$. For simplicity, we will write $[[Df]] = \sigma \|Df\|$. Note that $[[Df]]$ is a vector-valued Radon measure. When $f\in \L^1(\Omega)$ and $\|Df\|(\Omega)$ is finite, we say $f$ is of bounded variation on $\Omega$; that is $f\in BV(\Omega)$. The space of $BV(\Omega)$ is well studied and we refer the readers to the classical books \cite{MR3409135, MR1857292} for additional properties of $BV$ functions.

There have been several generalizations to the definition of $BV(\Omega)$ (and correspondingly $BV_{\loc}(\Omega)$). One direction of such generalization is that the Euclidean norm (applied to $T$) in \eqref{eq 101} can be replaced by any (potentially asymmetric) norms. Fix $L\in \mathcal{K}_o^n$, let $\|\cdot\|_{L}$ and $\|\cdot\|_{L^*}$ be as defined in \eqref{eq 210}. Note that since $L$ is compact and contains the origin in its interior, both $\|\cdot\|_L$ and $\|\cdot\|_{L^*}$ are equivalent to the standard Euclidean norm and therefore, the space $BV_L(\Omega)$ (and $BV_{L,\loc} (\Omega)$, resp.) consisting of $\L^1(\Omega)$ functions with
\begin{equation}
	TV_L(f; \Omega) = \sup\left\{\int_{\Omega} f\Div T\, dx: T\in C_c^1(\Omega, {\mathbb R^n}), \|T(x)\|_L\leq 1, \forall x \right\}<\infty,
\end{equation}
remains unchanged when compared to $BV(\Omega)$ (and $BV_{\loc}(\Omega)$, resp.). However, the anisotropic total variation $TV_L(f; \Omega)$ is generally not the same as $TV(f;\Omega)$. As a matter of fact, when $f=1_K$ for some $K\in \mathcal{K}^n$, then $TV(1_K;{\mathbb R^n})$ gives the surface area of $K$ whereas $TV_L(1_K; {\mathbb R^n})$ gives the mixed volume $V_1(K,L)=\int_{ {S^{n-1}}}h_L dS_K$. For $f\in BV_{\loc}(\Omega)$, we may define the anisotropic total variation measure with respect to $L$ by
\begin{equation}
	\|\shortminus Df\|_{L^*} = h_L(\shortminus \sigma) \|Df\|.
\end{equation}
It can be shown that $f\in BV_L(\Omega)$ if and only if $\|\shortminus Df\|_{L^*}$ is a finite measure. Moreover, we have $TV_L(f; \Omega)=\|\shortminus Df\|_{L^*}(\Omega)$. Functions of bounded anisotropic total variation were studied in for example, \cite{MR3055988, MR2672283} where anisotropic isoperimetric inequality and anisotropic Sobolev inequalities were studied for sets of finite perimeter and functions of bounded variation. It is important to notice that many of the classical results mentioned in the standard books \cite{MR3409135, MR1857292} work in the anisotropic setting with only very minor alterations to the proofs.

Another direction of generalization to $BV(\Omega)$ is to replace the Lebesgue measure in ${\mathbb R^n}$ by a generic measure. Things start to get complicated in this setting. As an example, the approximation of such $BV$ functions by smooth ones might fail. This explains why there are non-equivalent ways of defining $BV$ functions in a generic measure space $({\mathbb R^n}, \mu)$. We mention \cite{MR2005202} for one of the approaches for where $\mu$ is a doubling measure. When $\mu = \omega(x)dx$ is absolutely continuous with respect to Lebesgue measure, another way of generalizing the classical total variation (not equivalent to the one given in \cite{MR2005202}; see Section 5.1 in \cite{MR2005202}) was given in \cite{MR1864805}.

For our purpose, we adopt the following definition. We say a function $f\in \mathcal{L}^1(\omega)$ is of bounded anisotropic weighted variation (or $f\in BV_{L, \omega}(\Omega))$ if $f\in BV_{\loc}(\Omega)$ and $\omega\in \L^1(\|\shortminus Df\|_{L^*}, \Omega)$. In this case, we define the $(L,\omega)$-anisotropic total variation of $f$ to be
	\begin{equation}
	\label{eq 21}
		 TV_{L,\omega}(f;\Omega)=\int_{\Omega}\omega d\|\shortminus Df\|_{L^*}.
	\end{equation}
To see how this is connected to the classical definition \eqref{eq 101}, we mention that when $\omega: {\mathbb R^n} \rightarrow [0,\infty]$ is a lower semi-continuous function with $\omega(x)>0$ for all $x\neq o$, using an approximation argument in both $f$ and $\omega$, one can see that
\begin{equation}
	TV_{L, \omega}(f;\Omega)= \sup \left\{\int_{\Omega} f\Div T\, dx: T\in C_c^1(\Omega, {\mathbb R^n}), \|T(x)\|_L\leq \omega(x)\right\}.
\end{equation}
Since this representation is not required in the current work, we do not provide a proof here.

 Let $E\subset {\mathbb R^n}$ be a measurable set. When $1_E\in BV_{L,\omega}({\mathbb R^n})$, we say $E$ has finite $(L,\omega)$-anisotropic weighted perimeter and write $\Per_{L,\omega}(\partial E) =TV_{L,\omega}(1_E;{\mathbb R^n})$.

 Let $f:{\mathbb R^n}\rightarrow \mathbb{R}$ and $t\in \mathbb{R}$, define
 \begin{equation}
 	E_t=\{x\in {\mathbb R^n}: f(x)>t\}.
 \end{equation}

 For $BV$ functions, the following version of the classical coarea formula can be found in Figalli-Maggi-Pratelli \cite[(2.22)]{MR2672283}: if $f\in BV({\mathbb R^n})$ and $\zeta:{\mathbb R^n} \rightarrow [0,\infty]$ is a Borel function, then
 \begin{equation}
 	\int_{\mathbb R^n} \zeta d\|\shortminus Df\|_{L^*}  = \int_{-\infty}^\infty\left( \int_{{\mathbb R^n}} \zeta d\|\shortminus D1_{E_t}\|_{L^*}\right)dt.
 \end{equation}
 In particular, this implies
\begin{equation}
 \label{eq 26}
 	TV_{L,\omega}(f;{\mathbb R^n}) = \int_{-\infty}^\infty \Per_{L,\omega}(\partial E_t)dt.
 \end{equation}

 \section{Log-concave functions and their $(L,\omega)$ total variation}
 \label{section log concave variation formula}

  It is well-known that the set of convex bodies can be embedded naturally into $\Lc({\mathbb R^n})$ via their characteristic functions. Let $f = e^{-\phi}$, $g=e^{-\psi}$ be in $\Lc({\mathbb R^n})$ and $t>0$. The sup-convolution between $f$ and $g$ can be defined via the Legendre-Fenchel conjugate of their respectively associated convex functions:
\begin{equation}
	f\oplus t\cdot g= e^{-(\phi^*+t\psi^*)^*}.
\end{equation}
It is well-known that when $f=1_K$ and $g=1_L$ are characteristic functions of convex bodies, then
\begin{equation}
	1_K\oplus t\cdot 1_L = 1_{K+tL},
\end{equation}
where $K+tL$ is the usual Minkowski combination between convex bodies. Therefore, the supremum convolution $\oplus$ can be viewed as a natural generalization of Minkowski addition for convex bodies.

For each $q>0$. The $(q-n)$-th moment of a log-concave function $f$ is defined as
\begin{equation}
	\widetilde{V}_q(f) = \int_{\mathbb R^n} |x|^{q-n}f(x)dx.
\end{equation}
When $f=1_K$ for some convex body $K\in \mathcal K_o^n$, by using polar coordinates, it is simple to see that
\begin{equation}
	\widetilde{V}_q(1_K) = \frac{1}{q}\int_{ {S^{n-1}}}\rho_K^q(u)du = \widetilde{V}_q(K),
\end{equation}
where $\widetilde{V}_q(K)$ is the $q$-th dual quermassintegral of $K$. Therefore, the quantity $\widetilde{V}_q$ can be viewed as the natural generalization of dual quermassintegrals for convex bodies.

In the seminal work \cite{MR3573332, MR3783409}, the differentials of dual quermassintegrals were studied, which led to a family of long-sought-for geometric measures known as \emph{$(p,q)$-dual curvature measures}. These measures, since their discovery, have been the center of modern convex geometric analysis and their characterization problems (called \emph{Minkowski-type problems}) have been intensively studied in the past few years and have already led to many interesting conjectures regarding isoperimetric inequalities as well as the discovery of novel curvature flows.

It is natural to wonder whether the same philosophy can be applied in the space of log-concave functions---given that all the elements (dual quermassintegrals and Minkowski addition) have their natural counterparts in the larger space. It is the intention of the current section to demonstrate that the answer is yes, with some minor restrictions on the log-concave function $f$. To explain why these restrictions are needed, note that the variational formulas demonstrated in \cite{MR3573332, MR3783409}  require the convex body to contain the origin as an interior point and consequently the family of $(p,q)$-dual curvature measures are only defined for convex bodies that have the origin as an interior point. See, for example, equation (1.9) in \cite{MR3573332}. Therefore, it is natural to expect certain restrictions on $f$ that mimic the requirement that $K$ has origin in the interior as in the convex body case.

In the following, we show that the variation of the moment of $f$ is strongly connected to the theory of functions of bounded anisotropic weighted variation in Section \ref{sec BV}.

Throughout this section, if not specified otherwise, we let $q>0$.

We will study the existence of the following limit
\begin{equation}
\label{eq 33}
	\delta_q(f,g) = \lim_{t\rightarrow 0^+} \frac{\widetilde{V}_q(f\oplus t\cdot g)-\widetilde{V}_q(f)}{t}.
\end{equation}
Note that it is not clear at all why the limit should exist.

For the rest of the section, we write $\omega_q$ for the weight function
\begin{equation}
	\omega_q(x) = |x|^{q-n},
\end{equation}
for $q>0$. It is important to emphasize that the following proofs actually work for more general weight functions. As an example, the proofs (with very minor modifications) will work for Gaussian weight function $\omega(x)=e^{-|x|^2/2}$.



We require the following lemma from \cite{MR3573332}.
\begin{lemma}[\cite{MR3573332}]\label{lemma 122} Let $K\in \mathcal{K}_o^n$ and $L$ be a compact convex set in ${\mathbb R^n}$. Then, we have
\begin{equation}
\label{eq 35}
	\lim_{t\rightarrow 0^+}\frac{\widetilde{V}_q(K+tL)-\widetilde{V}_q(K)}{t} = \int_ {S^{n-1}} h_L(v)d\widetilde{C}_{1,q}(K,v),
\end{equation}	
where the definition of $\widetilde{C}_{1,q}(K,\cdot)$ is given in Section \ref{section preliminary convex body}.
\end{lemma}

We will denote the right-hand-side of \eqref{eq 35} by $\widetilde{V}_{1,q}(K,L)$; that is
\begin{equation}
	\widetilde{V}_{1,q}(K,L)= \int_ {S^{n-1}} h_L(v)d\widetilde{C}_{1,q}(K,v)= \int_ {S^{n-1}} \frac{h_L(v)}{h_K(v)}d\widetilde{C}_{q}(K,v).
\end{equation}

It was shown in Rotem \cite{https://doi.org/10.48550/arxiv.2006.16933} that if $f\in \Lc({\mathbb R^n})$ and $f$ is $\L^1$, then $f \in BV({\mathbb R^n})$ and
\begin{equation}
\label{eq 32}
	[[Df]] = \nabla f\,dx-f\nu_{\partial K_f} d\mathcal{H}^{n-1}|_{\partial K_f},
\end{equation}
where $K_f$ is the support of $f$ and is thus convex by the fact that $f\in \Lc({\mathbb R^n})$. With this in mind, by \eqref{eq 21}, it is simple to see that if $K\in \mathcal{K}_o^n$, then $K$ is of finite anisotropic weighted perimeter. Indeed,
\begin{equation}
\label{eq 31}
\begin{aligned}
	\Per_{L,\omega_q}(\partial K) &= TV_{L,\omega_q}(1_K, {\mathbb R^n}) \\
	&= \int_{\partial K} h_L(\nu_{K}(x))|x|^{q-n}dx \\
	&= \int_{ {S^{n-1}}} h_L(v)d\widetilde{C}_{1,q}(K,v)\\
	&=\widetilde{V}_{1,q}(K,L)<\infty.
\end{aligned}
\end{equation}

We also require the following lemmas for convex bodies.
\begin{lemma}
\label{lemma a11}
 	Let $K \in \mathcal{K}_o^n$ and $r_0>0$ be such that
 	\begin{equation}
 		B(r_0)\subset K.
 	\end{equation}
 	Let $L\in \mathcal{K}^n$ and denote
 	\begin{equation}
 		K_t = K+tL.
 	\end{equation}
 	Then, there exists $C>0, \delta_0>0$ which depends only on $r_0$ and $\max_{L}|x|$, such that
 	\begin{equation}
 		\left|\frac{\log \rho_{K_t}-\log \rho_K}{t}\right|<C,
 	\end{equation}
 	for every $|t|<\delta_0$.
 \end{lemma}

 \begin{proof}
 	Note that
 	\begin{equation}
 	\begin{aligned}
 		\log h_{K_t} &= \log (h_K+th_L) \\
 		&= \log h_K + \log(1+th_L/h_K) \\
 		&= \log h_K+ t\frac{h_L}{h_K} + o(t,\cdot),
 	\end{aligned}
 	\end{equation}
 	where
 	\begin{equation}
 		|o(t,v)| \leq \frac{h_L^2}{2(h_K-|th_L|)^2}t^2.
 	\end{equation}
 	It is simple to see that there exists $\delta_0, C_1>0$ that only depends on $r_0$ and $\max_{L}|x|$, such that for each $|t|<\delta_0$, we have
 	\begin{equation}
 		\left|\frac{o(t, \cdot)}{t}\right|<C_1,
 	\end{equation}
 	uniformly in $t$ and $v$. According to (4.12) and Lemma 2.8 in \cite{MR3573332}, we have
 	\begin{equation}
 		\left|\frac{\log \rho_{K_t}-\log \rho_K}{t}\right|\leq \max \frac{h_L}{h_K}+\max \left|\frac{o(t, \cdot)}{t}\right|<C,
 	\end{equation}
 	where $C>0$ only depends on $r_0$, and $\max_{L}|x|$.
 \end{proof}

 \begin{coro}
 \label{corollary 11}
 	Under the same assumptions in Lemma \ref{lemma a11}, for each $q>0$, we have
 	\begin{equation}
 		\left|\frac{\rho^q_{K_t}-\rho^q_K}{t}\right|<2^qCq\rho_K^q.
 	\end{equation}
 \end{coro}
 \begin{proof}
 By mean value theorem and Lemma \ref{lemma a11},
 	\begin{equation}
 		\begin{aligned}
 			\left|\frac{\rho^q_{K_t}-\rho^q_K}{t}\right|&= \left|\frac{e^{q\log \rho_{K_t}}-e^{q\log \rho_{K}}}{\log \rho_{K_t}-\log \rho_{K}}\right|\left|\frac{\log \rho_{K_t}-\log \rho_K}{t}\right|\\
 			&\leq C\left|\frac{e^{q\log \rho_{K_t}}-e^{q\log \rho_{K}}}{\log \rho_{K_t}-\log \rho_{K}}\right|\\
 			& = Cq\theta^q,
 		\end{aligned}
 	\end{equation}
 	where $\theta$ is between $\rho_{K_t}$ and $\rho_K$. Note that since $K$ is bounded from below and $L$ is bounded from above, for sufficiently small $\delta_0$ (dependent on $r_0$ and $\max_L |x|)$, we have $\theta<2\rho_K$, which immediately gives the desired bound.
 \end{proof}

 \begin{theorem}
  \label{theorem delta layer cake}
 	Let $f\in \Lc({\mathbb R^n})$ with non-zero finite $\L^1$ norm and $L\in \mathcal{K}_o^n$. Assume $f$ achieves its maximum at $o$ and
 		\begin{equation}
 		\label{eq 400}
 			\limsup_{x\rightarrow o} \frac{|f(x)-f(o)|}{|x|^{\alpha+1}}<\infty,
 		\end{equation}
 		for some $0<\alpha<1$.
 	Then,
 	\begin{equation}
 		\delta_q (f, 1_L)= \int_{0}^\infty \widetilde{V}_{1,q}([f\geq s],L)\,ds<\infty.
 	\end{equation}
 \end{theorem}

 \begin{proof}
 	
 	For simplicity, write
 	\begin{equation}
 		K_s = [f\geq s],
 	\end{equation}
 	and $M=f(o)=\max f$.
 	By assumption, there exists $\varepsilon_0>0$ and $c_0>0$ such that
 	\begin{equation}
 		K_s\supset c_0(M-s)^{\frac{1}{\alpha+1}} B,
 	\end{equation}
 	for any $s\in (M-\varepsilon_0, M)$. In particular, this implies, for each $s<M$, the set $K_s$ contains the origin in the interior. We also require that $\varepsilon_0$ is sufficiently small so that $K_s\subset c_1B$ for each $s\in (M-\varepsilon_0, M)$ and some $c_1>0$. This is possible because the fact that $f\in \L^1$ implies that $\lim_{|x|\rightarrow \infty}f(x)=0$.
 	
 	Note that by layer cake representation, we have
 	\begin{equation}
 		\delta_q (f, 1_L) = \lim_{t\rightarrow 0^+} \int_{0}^M\frac{ \widetilde{V}_q(K_s+tL)-\widetilde{V}_q(K_s)}{t}ds
 	\end{equation}
 	
 	Step 1:
 	\begin{equation}
 		\lim_{t\rightarrow 0^+} \int_{M-\varepsilon_0}^{M}\frac{ \widetilde{V}_q(K_s+tL)-\widetilde{V}_q(K_s)}{t}ds=\int_{M-\varepsilon_0}^{M}\lim_{t\rightarrow 0^+}\frac{ \widetilde{V}_q(K_s+tL)-\widetilde{V}_q(K_s)}{t}ds.
 	\end{equation}
 	\begin{proof}[Proof of step 1:]
 		Let
 		\begin{equation}
 			g(t;s)=\widetilde{V}_q(K_s+tL).
 		\end{equation}
 		By Lemma \ref{lemma 122}, for each $s\in (M-\varepsilon_0,M)$, the function $g(t;s)$ is differentiable in $t$ and
 		\begin{equation}
 			\frac{\partial}{\partial t} g(t;s) = \int_{ {S^{n-1}}} \frac{h_L(v)}{h_{K_s+tL}(v)}d\widetilde{C}_q(K_s+tL,v).
 		\end{equation}
 		Therefore, by mean value theorem
 		\begin{equation}
 			\frac{g(t;s)-g(0;s)}{t} = \int_{ {S^{n-1}}} \frac{h_L(v)}{h_{K_s+\theta L}(v)}d\widetilde{C}_q(K_s+\theta L,v),
 		\end{equation}
 		where $\theta\in [0,t]$ and is dependent on $s$. By the fact that $L$ is compact, and $K_s+\theta L\supset K_s\supset c_0(M-s)^{\frac{1}{1+\alpha}} B$, we have
 		\begin{equation}
 		\begin{aligned}
 			\frac{g(t;s)-g(0;s)}{t}&\leq \max_{L}|x|\cdot \frac{1}{c_0}(M-s)^{-\frac{1}{1+\alpha}}\widetilde{V}_q(K_s+L)\\
 			&\leq \max_{L}|x|\cdot\frac{1}{c_0}(M-s)^{-\frac{1}{1+\alpha}}\widetilde{V}_q(c_1B+L)\\
 			&\leq C(M-s)^{-\frac{1}{1+\alpha}},
 		\end{aligned}	
 		\end{equation}
 		for some $c_1>0$.
 		
 		Note that $(M-s)^{-\frac{1}{1+\alpha}}$ is integrable near $M$ thanks to $\alpha>0$. Therefore, by dominated convergence theorem, we get the desired result.
 	\end{proof}
 	Step 2:
 	\begin{equation}
 		\lim_{t\rightarrow 0^+} \int_0^{M-\varepsilon_0}\frac{ \widetilde{V}_q(K_s+tL)-\widetilde{V}_q(K_s)}{t}ds=\int_0^{M-\varepsilon_0}\lim_{t\rightarrow 0^+}\frac{ \widetilde{V}_q(K_s+tL)-\widetilde{V}_q(K_s)}{t}ds.
 	\end{equation}
 	\begin{proof}[Proof to step 2:]
		For each $s\in (0,M-\varepsilon_0)$, there exists $\lambda_0>0$ such that
		\begin{equation}
			K_s\supset K_{M-\varepsilon_0}\supset \lambda_0B.
		\end{equation} 	
		
	By Corollary \ref{corollary 11}, for sufficiently small $|t|$,
	\begin{equation}
		\begin{aligned}
			\frac{ \widetilde{V}_q(K_s+tL)-\widetilde{V}_q(K_s)}{t}= \int_{ {S^{n-1}}} \frac{\rho^q_{K_s+tL}-\rho_{K_s}^q}{t}du\leq 2^qCq\int_{ {S^{n-1}}} \rho_{K_s}^qdu,
		\end{aligned}
	\end{equation}
	where the constant $C$ here depends on $\lambda_0$ and $\max_{L}|x|$. In particular, $C$ is not dependent on $s$.
	
	Note now that
	\begin{equation}
		\frac{1}{q}\int_{0}^{M-\varepsilon_0}\int_{ {S^{n-1}}} \rho_{K_s}^qdu ds= \int_{0}^{M-\varepsilon_0}\widetilde{V}_q(K_s)ds\leq \int_{0}^M\widetilde{V}_q(K_s)ds = \int_{{\mathbb R^n}}|x|^{q-n}f(x)dx<\infty.
	\end{equation}
	
	Therefore, we may use dominated convergence theorem to justify the exchange of limit and integration.
 	\end{proof}
 	
 	By Step 2 and Step 3, we have
 	\begin{equation}
 		\begin{aligned}
 			\delta_q (f, 1_L) &=\int_{0}^M \lim_{t\rightarrow 0^+}  \frac{ \widetilde{V}_q(K_s+tL)-\widetilde{V}_q(K_s)}{t}ds\\
 			&=\int_{0}^M \widetilde{V}_{1,q}([f\geq s],L)ds\\
 			&=\int_0^\infty \widetilde{V}_{1,q}([f\geq s],L)ds.
 		\end{aligned}
 	\end{equation}
 	
 	Note that it is included in the proof of Steps 1 and 2 that the right-hand-side is finite.
 \end{proof}

 We first show the validity of Theorem \ref{theorem intro 1} in the special case that $g$ is a constant multiple of a characteristic function.
 \begin{theorem}
 \label{theorem 11}
 		Let $f\in \Lc({\mathbb R^n})$ with non-zero finite $\L^1$ norm and $L\in \mathcal{K}^n$. Assume $f$ achieves its maximum at $o$ and
 		\begin{equation}
 		\label{eq 400aa}
 			\limsup_{x\rightarrow o} \frac{|f(x)-f(o)|}{|x|^{\alpha+1}}<\infty,
 		\end{equation}
 		for some $0<\alpha<1$.
 		
 		Let $g=c1_L=e^{-\psi}$ for some $c>0$. Then,
 		\begin{equation}
 			\delta_q(f,g) = \int_{\mathbb R^n} \psi^*(y)d\widetilde{C}^{e}_{q}(f;y)+ \int_{ {S^{n-1}}} h_L(v)d\widetilde{C}_q^s(f;v).
 		\end{equation}
 \end{theorem}
 \begin{proof}
 	We first restrict our attention to the case where $L\in \mathcal{K}_o^n$.

 	We may assume without loss of generality that $c=1$. In this case, $\psi^*=h_L$.
 	
 	According to Theorem \ref{theorem delta layer cake}, \eqref{eq 31}, and \eqref{eq 26},  we have
 	\begin{equation}
 		\begin{aligned}
 			\infty>\delta_q (f, 1_L)&= \int_{0}^\infty \widetilde{V}_{1,q}([f\geq s],L)\,ds\\
 			&=\int_{0}^{\infty} \Per_{L,\omega_q}(\partial [f\geq s])ds\\
 			&=TV_{L,\omega_q}(f;{\mathbb R^n}).
 		\end{aligned}
 	\end{equation}
 	By \eqref{eq 21} and \eqref{eq 32}, we have
 	\begin{equation}
 	\begin{aligned}
 		\delta_q (f, 1_L)=TV_{L,\omega_q}(f) &= \int_{\mathbb R^n} h_L(\nabla \phi)f(x)|x|^{q-n}dx+ \int_{\partial K_f}h_L(\nu_{\partial K_f})f(x)|x|^{q-n}d\mathcal{H}^{n-1}(x)\\
 		&=\int_{\mathbb R^n} h_L(y)d\widetilde{C}^{e}_{q}(f;y)+\int_ {S^{n-1}} h_L(v)\widetilde{C}_q^s(f;v).
 	\end{aligned}
 	\end{equation}
 	Here, the last equality follows straightly from the definition of $\widetilde{C}^{e}_{q}$ and $\widetilde{C}_q^s$. See Definitions \ref{def intro 1} and \ref{def intro 2}.
 	By linearity of the above equation in $L$, we can remove the restriction that $L$ contains the origin in its interior.
\end{proof}

We need the following lemma from \cite{https://doi.org/10.48550/arxiv.2206.13146}.
\begin{lemma}
  \label{lemma pointwise limit}
  	Let $f=e^{-\phi}, g=e^{-\psi}\in \Lc({\mathbb R^n})$ be such that $f$ has nonzero finite $\L^1$ norm and $g$ is compactly supported. Then for almost all $x\in {\mathbb R^n}$, we have
  	\begin{equation}
  		\lim_{t\rightarrow 0^+} \frac{(f\oplus(t\cdot g))(x)-f(x)}{t} = \psi^*(\nabla \phi(x))f(x).
  	\end{equation}
  \end{lemma}

We are now ready to prove the promised variational formula; that is, compute the limit in \eqref{eq 33}.
\begin{theorem}
\label{theorem 31}
 		Let $f=e^{-\phi}\in \Lc({\mathbb R^n})$ with non-zero finite $\L^1$ norm. Assume $f$ achieves its maximum at $o$ and
 		\begin{equation}
 		\label{eq 400bb}
 			\limsup_{x\rightarrow o} \frac{|f(x)-f(o)|}{|x|^{\alpha+1}}<\infty,
 		\end{equation}
 		for some $0<\alpha<1$. Let $g=e^{-\psi}\in \Lc({\mathbb R^n})$ be compactly supported. Then,
 		\begin{equation}
 		\label{eq 40}
 			\delta_q(f,g) = \int_{\mathbb R^n} \psi^*(y)d\widetilde{C}^{e}_{q}(f;y)+ \int_{ {S^{n-1}}} h_{K_g}(v)d\widetilde{C}_q^s(f;v).
 		\end{equation}
 \end{theorem}
 \begin{proof}
 	Let $f_t = f\oplus(t\cdot g)$. Since $g$ is compactly supported, there exists $A>0$ such that $g\leq A 1_{K_g}$ and therefore $f_t\leq \widetilde{f}_t= f\oplus(t\cdot A1_{K_g})$.
 	
 	By Lemma \ref{lemma pointwise limit},
 	\begin{equation}
 		\lim_{t\rightarrow 0^+} \frac{\widetilde{f}_t-f_t}{t} = \lim_{t\rightarrow 0^+} \frac{\widetilde{f}_t-f}{t}-\lim_{t\rightarrow 0^+} \frac{{f}_t-f}{t} = h_{K_g}(\nabla \phi)f+\ln A f-\psi^*(\nabla \phi)f.
 	\end{equation}
 	
 	By Fatou's lemma, we have
 	\begin{equation}
 	\begin{aligned}
 		\liminf_{t\rightarrow 0^+}\int_{\mathbb R^n} \frac{\widetilde{f}_t-f_t}{t}|x|^{q-n} dx&\geq \int_{\mathbb R^n} [h_{K_g}(\nabla \phi)f+\ln A f-\psi^*(\nabla \phi)f ]\cdot |x|^{q-n} dx\\
 		&=\int_{\mathbb R^n} (h_{K_g}(y)-\psi^*(y))d\widetilde{C}^{e}_{q}(f;y)+\ln A \cdot \widetilde{V}_q(f)
 	\end{aligned}
 	\end{equation}
 	
 	By Theorem \ref{theorem 11}, we have
 	\begin{equation}
 	\begin{aligned}
 			&\lim_{t\rightarrow 0^+}\int \frac{\widetilde{f}_t-f}{t}|x|^{q-n} dx \\
 			&= \int_{\mathbb R^n} h_{K_g}(y)d\widetilde{C}^{e}_{q}(f;y)+\ln A \cdot \widetilde{V}_q(f)+ \int_{ {S^{n-1}}} h_{K_g}(v)d\widetilde{C}_q^s(f;v).
 	\end{aligned}
 	\end{equation}
 	Combining the above two formulas, we have
 	\begin{equation}
 		\limsup_{t\rightarrow 0^+}\int \frac{f_t-f}{t}|x|^{q-n}dx\leq \int_{{\mathbb R^n}} \psi^*(y)d\widetilde{C}^{e}_{q}(f;y)+\int_{ {S^{n-1}}} h_{K_g}(v)d\widetilde{C}_q^s(f;v).
 	\end{equation}
 	
 	For the other direction of the inequality, define for each positive integer $j$, the set
 	\begin{equation}
 		Q_j = \left\{x\in {\mathbb R^n}: g(x)\geq \frac{1}{j}\right\}.
 	\end{equation}
 	Since $g$ is compactly supported, we conclude that $Q_j\uparrow K_g$. Let
 	\begin{equation}
 		\bar{g} = \frac{1}{j}1_{Q_j}, \quad \text{and } \bar{f}_t = f\oplus t\cdot \bar{g}.
 	\end{equation}
 	Note that $\bar{f}_t\leq f_t$. Argue the same way as before, we have
 	\begin{equation}
 		\liminf_{t\rightarrow 0^+}\int \frac{f_t-\bar{f}_t}{t}|x|^{q-n} dx\geq \int_{\mathbb R^n} (\psi^*(y)-h_{Q_j}(y))\,d\widetilde{C}^{e}_{q}(f;y)+\ln j \cdot \widetilde{V}_q(f),
 	\end{equation}
 	and
 	\begin{equation}
 	\begin{aligned}
 			&\lim_{t\rightarrow 0^+}\int \frac{\bar{f}_t-f}{t}|x|^{q-n} dx \\
 			&= \int_{\mathbb R^n} h_{Q_j}(y)d\widetilde{C}^{e}_{q}(f;y)-\ln j \cdot \widetilde{V}_q(f)+ \int_{ {S^{n-1}}} h_{Q_j}(v)d\widetilde{C}_q^s(v).
 	\end{aligned}
 	\end{equation}
 	Adding the above two formulas, we have
 	\begin{equation}
 		\liminf_{t\rightarrow 0^+}\int \frac{f_t-f}{t}|x|^{q-n}dx\geq \int_{\mathbb R^n} \psi^*(y)\,d\widetilde{C}^{e}_{q}(f;y)+ \int_{ {S^{n-1}}} h_{Q_j}(v)d\widetilde{C}_q^s(v).
 	\end{equation}
 	Letting $j\rightarrow \infty$, by monotone convergence theorem, we have
 	\begin{equation}
 		\liminf_{t\rightarrow 0^+}\int \frac{f_t-f}{t}|x|^{q-n}dx\geq \int_{\mathbb R^n} \psi^*(y)\,d\widetilde{C}^{e}_{q}(f;y)+ \int_{ {S^{n-1}}} h_{K_g}(v)d\widetilde{C}_q^s(v).
 	\end{equation}
 	This completes the proof.
 \end{proof}

 {
 We remark here that the hypothesis \eqref{eq 400} in Theorem \ref{theorem delta layer cake} (and consequently in Theorem \ref{theorem 31}) is not sharp. In particular, we show that if the level sets of $f$ near the origin are \emph{uniformly} in ``good shape'', then Theorem \ref{theorem delta layer cake} still holds.

For $K\in \mathcal{K}_o^n$, define
\begin{equation}
	r_K = \max\{r\geq 0: rB\subset K\}
\end{equation}
and
\begin{equation}
	R_K = \min\{r\geq 0: K\subset rB\}.
\end{equation}
\begin{prop}
\label{prop 8221}
	Let $f\in \Lc({\mathbb R^n})$ with non-zero finite $\L^1$ norm and $L\in \mathcal{K}_o^n$. Assume $f$ achieves its maximum at $o$. If there exists $\varepsilon_0>0$ and $C>0$ such that for almost all  $f(o)-\varepsilon_0<s<f(o)$, we have
	\begin{equation}
	\label{eq 8222}
		1\leq \frac{R_{[f\geq s]}}{r_{[f\geq s]}}<C,
	\end{equation}
	 	then
 	\begin{equation}
 		\delta_q (f, 1_L)= \int_{0}^\infty \widetilde{V}_{1,q}([f\geq s],L)\,ds<\infty.
 	\end{equation}
\end{prop}

\begin{proof}
	Denote $f(o)$ by $M$ and $K_s=[f\geq s]$. Like in the proof of Theorem \ref{theorem delta layer cake}, it is sufficiently to show
	\begin{equation}
	\label{eq 8221}
 		\lim_{t\rightarrow 0^+} \int_{M-\varepsilon_0}^{M}\frac{ \widetilde{V}_q(K_s+tL)-\widetilde{V}_q(K_s)}{t}ds=\int_{M-\varepsilon_0}^{M}\lim_{t\rightarrow 0^+}\frac{ \widetilde{V}_q(K_s+tL)-\widetilde{V}_q(K_s)}{t}ds,
 	\end{equation}
 	and
 	\begin{equation}
 	\label{eq 8225}
 		\lim_{t\rightarrow 0^+} \int_0^{M-\varepsilon_0}\frac{ \widetilde{V}_q(K_s+tL)-\widetilde{V}_q(K_s)}{t}ds=\int_0^{M-\varepsilon_0}\lim_{t\rightarrow 0^+}\frac{ \widetilde{V}_q(K_s+tL)-\widetilde{V}_q(K_s)}{t}ds.
 	\end{equation}
 	Note that the latter follows in the same way as before. Hence, we only need to justify \eqref{eq 8221}.
 	
 	By \eqref{eq 8222}, it is simple to see that for almost all $s\in (M-\varepsilon_0, M)$ and $t\in (0,1)$, we have
 	\begin{equation}
 		\frac{R_{K_s+tL}}{r_{K_s+tL}}<C_1
 	\end{equation}
 	for some $C_1>0$ independent of $s$ and $t$. Consequently, there exists $c_2>0$ independent of $s$ and $t$ such that
 	\begin{equation}
 	\label{eq 8223}
 		c_2<\nu_{K_s+tL}(x)\cdot \frac{x}{|x|}\leq 1,
 	\end{equation}
 	for almost all $x\in \partial (K_s+tL)$.
 	
 	Let $g(t;s)$ be as defined in the proof of Theorem \ref{theorem delta layer cake}. By the same argument, we have
 	\begin{equation}
 			\frac{g(t;s)-g(0;s)}{t} = \int_{ {S^{n-1}}} \frac{h_L(v)}{h_{K_s+\theta L}(v)}d\widetilde{C}_q(K_s+\theta L,v),
 		\end{equation}
 		where $\theta\in [0,t]$ and is dependent on $s$. By the fact that $L$ is compact and \eqref{eq 8223}, there exists $C_3>0$ independent of $t$ and $s$ such that
 		\begin{equation}
 			\frac{g(t;s)-g(0;s)}{t}\leq C_3\widetilde{V}_{q-1}(K_s+\theta L).
 		\end{equation}
 		
 		When $q\geq 1$, using the monotonicity of $\widetilde{V}_{q-1}$, we have
 		\begin{equation}
 			\frac{g(t;s)-g(0;s)}{t}\leq C_3 \widetilde{V}_{q-1}(K_{M-\varepsilon_0}+L)\leq C_4,
 		\end{equation}
 		for some $C_4>0$. Equation \eqref{eq 8221} then follows from bounded convergence theorem.
 		
 		Let us now concentrate on the case $q\in (0,1)$. By the fact that $f$ is finite in a neighborhood of the origin, there exists $c_5>0$ such that
 		\begin{equation}
 			f(x)\geq M/2,
 		\end{equation} 	
 		for all $|x|\leq c_5$. By log-concavity of $f$, we have
 		\begin{equation}
 		\begin{aligned}
 			f(y) &= f\left(\left(1-\frac{|y|}{c_5}\right)o+ \frac{|y|}{c_5}\left(\frac{y}{|y|}c_5\right)\right)\\
 			&\geq M^{1-\frac{|y|}{c_5}}\left(\frac{M}{2}\right)^{\frac{|y|}{c_5}}\\
 			&= M \cdot 2^{-\frac{|y|}{c_5}},
 		\end{aligned}		
 		\end{equation}	
 		for every $|y|\leq c_5$. Consequently, there exists $\delta_0>0$ and $C_6>0$  such that for all $s\in (M-\delta_0,M)$, we have
 		\begin{equation}
 			K_s \supset C_6(\log M-\log s)B.
 		\end{equation}
 		Therefore, by monotonicity of $\widetilde{V}_{q-1}$, for each $s\in (M-\delta_0,M)$, we have
 		\begin{equation}
 			\frac{g(t;s)-g(0;s)}{t}\leq C_3 \widetilde{V}_{q-1}(K_{s})\leq C_7 (\log M-\log s)^{q-1},
 		\end{equation}
 		for some $C_7>0$. Note that
 		\begin{equation}
 			\int_{M-\delta_0}^M(\log M-\log s)^{q-1}ds<\infty.
 		\end{equation}
 		Hence, by dominated convergence theorem, we conclude the validity of \eqref{eq 8221} with $\varepsilon_0$ replaced by $\delta_0$. Note that \eqref{eq 8225} holds with $\varepsilon_0$ replaced by $\delta_0$ as well---via the exact same argument. Therefore, we conclude the desired result.
 		 \end{proof}
 		
 \begin{remark}
 	We remark here that since in the proof of Theorem \ref{theorem 31} we required \eqref{eq 400bb} only for the ability to apply Theorem \ref{theorem delta layer cake}, by Proposition \ref{prop 8221}, we conclude that Theorem \ref{theorem 31} continues to hold with \eqref{eq 400bb} replaced by \eqref{eq 8222}. In particular, hypothesis \eqref{eq 8222} allows for log-concave functions such as
 	\begin{equation}
 		f(x)=e^{-\|x\|_K},
 	\end{equation}
 	where $K\in\mathcal{K}^{n}_o$. It is of great interest to see whether assumptions like \eqref{eq 400bb} and \eqref{eq 8222} can be dropped altogether.
 \end{remark}}

A few additional remarks are in order:
\begin{enumerate}
	\item Theorem \ref{theorem 31} justifies why we referred to $\widetilde{C}^{e}_{q}(f;\cdot)$ and $\widetilde{C}_q^s(f;\dot)$ as the Euclidean and spherical dual curvature measures for log-concave functions. In particular, when $f$ and $g$ are characteristic functions of convex bodies containing the origin in their respective interiors, \eqref{eq 40} recovers its convex geometric counterpart \eqref{eq 35}.
	\item Since $\phi$ is convex and therefore almost everywhere differentiable in the support of $f$, the measure $\widetilde{C}^{e}_{q}(f;\cdot)$ is well-defined. Notice that its total measure is equal to $\widetilde{V}_q(f)$. Therefore, by Proposition \ref{prop 12}, as long as $f$ is in $\L^1$, the measure $\widetilde{C}^{e}_{q}(f;\cdot)$ is always finite.
	\item Since $f$ is log-concave, its support $K_f$ is necessarily convex and therefore it makes sense to write $\nu_{\partial K_f}$. However, even with the assumption that $f\in \L^1$, it might not be the case that the measure $\widetilde{C}^{s}_{q}(f;\cdot)$ is finite. To see this, simply take the example where $K_f$ has a facet that contains the origin in the relative interior. In that case, the density $|x|^{q-n}$ will have a non-integrable singularity at the origin in that subspace when $q<1$. This counterexample suggests that we must impose some condition on $f$ such that $f$ ``contains the origin in the interior''. With the additional assumption that $K_f$ contains the origin in its interior, one may show that $\widetilde{C}^{s}_{q}(f;\cdot)$ is always a finite measure.
	\mycomment{
	\begin{proof}
		We provide a proof here for self-reference. Since $f\in \L^1$, there exists $A>0$ and $c>0$ such that
		\begin{equation}
			f(y)\leq Ae^{-c|y|}.
		\end{equation}
		
		 Note that
		\begin{equation}
			e^{-c|x|} = c\int_0^\infty e^{-ct}1_{tB}(x)dt.
		\end{equation}
		
		Case 1: $q\in (0,n]$.
		Since $K_f$ contains the origin in its interior, we have $r_0B\subset K_f$ for some $r_0>0$. Then
		\begin{equation}
			\begin{aligned}
				\int_{\partial K_f} |y|^{q-n}f(y)d\mathcal{H}^{n-1}(y)&\leq Ac \int_{0}^\infty\int_{\partial K_f} e^{-ct}1_{tB}(x)|x|^{q-n}d\mathcal{H}^{n-1}(x)dt\\
				&=Ac \int_{0}^\infty e^{-ct}\int_{\partial K_f\cap tB}|x|^{q-n}d\mathcal{H}^{n-1}(x)dt\\
				&\leq Acr_0^{q-n}\int_0^\infty e^{-ct}\int_{\partial K_f\cap tB}d\mathcal{H}^{n-1}(x)dt\\
				&\leq Ac r_0^{q-n}\int_{0}^\infty e^{-ct}\int_{\partial (K_f\cap tB)}d\mathcal{H}^{n-1}(x)dt\\
				&\leq Ac r_0^{q-n}\int_{0}^\infty e^{-ct}\int_{\partial (tB)}d\mathcal{H}^{n-1}(x)dt<\infty.
			\end{aligned}
		\end{equation}
		
		Case 2: $q>n$.
		\begin{equation}
			\begin{aligned}
				\int_{\partial K_f} |y|^{q-n}f(y)d\mathcal{H}^{n-1}(y)&\leq Ac \int_{0}^\infty\int_{\partial K_f} e^{-ct}1_{tB}(x)|x|^{q-n}d\mathcal{H}^{n-1}(x)dt\\
				&=Ac \int_{0}^\infty e^{-ct}\int_{\partial K_f\cap tB}|x|^{q-n}d\mathcal{H}^{n-1}(x)dt\\
				&\leq Ac\int_0^\infty e^{-ct}|t|^{q-n}\int_{\partial K_f\cap tB}d\mathcal{H}^{n-1}(x)dt\\
				&\leq Ac \int_{0}^\infty e^{-ct}|t|^{q-n}\int_{\partial (K_f\cap tB)}d\mathcal{H}^{n-1}(x)dt\\
				&\leq Ac \int_{0}^\infty e^{-ct}|t|^{q-n}\int_{\partial (tB)}d\mathcal{H}^{n-1}(x)dt<\infty.
			\end{aligned}
		\end{equation}
		
	\end{proof}}
	\item When $f=1_K$ is the characteristic function of a convex body $K\in \mathcal{K}_o^n$, the Euclidean dual curvature measure $\widetilde{C}^{e}_{q}(f;\cdot)$ reduces a point mass concentrated at the origin  and
	\begin{equation}
		\widetilde{C}^{s}_{q}(f;\cdot) = \widetilde{C}_{1,q}(K,\cdot).
	\end{equation}
	\item Let us emphasize again although the weight function $\omega_q=|x|^{q-n}$ is the only type of weight functions treated in this section, many results presented here can be shown for other weight functions (with only small changes). In particular, the Gaussian weight function is one of the many weight functions that can be used here to replace $\omega_q$. We mention that in the setting of convex bodies, variational formula for the Gaussian volume and its corresponding Gaussian Minkowski problem was previously studied by the authors in \cite{MR4252759}.
\end{enumerate}

Minkowski problems for the $(p,q)$-dual curvature measures in the setting of convex bodies have been the central topic in recent years. See, \cite{MR3944247,MR3953117,MR4127893,MR4259871,MR4271790}.
In particular, in the symmetric setting, the $(1,q)$-dual Minkowski problem have been completely solved in \cite{MR3953117,MR3810248}. In this paper, we shall focus on the characterization problem for the Euclidean dual curvature measure $\widetilde{C}^{e}_{q}(f;\cdot)$. This will be in the next section.

\section{The Minkowski problem for $\widetilde{C}^{e}_{q}$}

The purpose of this section is to study the following characterization problem for Euclidean curvature measure of log-concave functions.

\textbf{The functional dual Minkowski problem.} Let $q>0$ and $\mu$ be a non-zero finite Borel measure on ${\mathbb R^n}$. Find the necessary and sufficient condition on $\mu$ so that there exists $f\in \Lc({\mathbb R^n})$ with nonzero finite $\L^1$ norm such that
\begin{equation}
\label{eq local 2}
	\mu = \widetilde{C}^{e}_{q}(f;\cdot).
\end{equation}


Under sufficient regularity assumption; that is, the measure $\mu$ has a $C^\infty$ density (say, $g\geq 0$) and $f\in C^{\infty}$, \eqref{eq local 2} is equivalent to the following Monge-Amp\`{e}re type equation in ${\mathbb R^n}$
\begin{equation}
\label{eq local 3}
	g(\nabla \phi(x))\det (\nabla^2 \phi(x)) = |x|^{q-n}e^{-\phi(x)},
\end{equation}
where $f=e^{-\phi}$.

We will restrict our attention to the even case---the prescribed measure $\mu$ in  \eqref{eq local 2} is even and we are restricting our solution set to all even functions $f\in \Lc({\mathbb R^n})$.

\subsection{The variational structure} In this subsection, we convert the solvability of \eqref{eq local 2} into the existence of minimizer to a minimization problem.

We will require the following variational formula.
\begin{lemma}
\label{lemma 51}
	Let $f\in \Lc({\mathbb R^n})$ be an even function and $q>0$. Suppose $f$ has non-zero finite $(q-n)$-th moment and takes the form $f=e^{-\phi}$ for some $\phi\in \cvx({\mathbb R^n})$. For each $\zeta\in C_c({\mathbb R^n})$, define
	\begin{equation}
		\phi_t(x) = (\phi^*+t\zeta)^*(x),
	\end{equation}
	and
	\begin{equation}
		f_t(x)=e^{-\phi_t(x)}.
	\end{equation}
	Then, we have
	\begin{equation}
	\label{eq local 1}
		\left.\frac{d}{dt}\right|_{t=0}\widetilde{V}_q(f_t) = \int_{\mathbb R^n} \zeta(\nabla \phi(x))|x|^{q-n} f(x)\,dx = \int_{\mathbb R^n} \zeta(y)d\widetilde{C}^{e}_{q}(f;y).
	\end{equation}
\end{lemma}
\begin{proof}
	We note that the second equality follows directly from the definition of $\widetilde{C}^{e}_{q}(f;\cdot)$ and therefore only the first equality needs a proof.
	
	Recall that from Proposition 2.1 in \cite{https://doi.org/10.48550/arxiv.2006.16933}, we have
	\begin{equation}
		\left.\frac{\partial \phi_t(x)}{\partial t}\right|_{t=0} = -\zeta(\nabla \phi(x)),
	\end{equation}
	almost everywhere in ${\mathbb R^n}$.
	
	For simplicity, write
	\begin{equation}
		h_t(x) = |x|^{q-n} f_t(x).
	\end{equation}
	Since $\zeta\in C_c({\mathbb R^n})$, there exists $M>0$ such that $|\zeta|\leq M$. Therefore, we have
	\begin{equation}
		\phi^*-|t|M\leq \phi^*+t\zeta\leq \phi^*+|t|M.
	\end{equation}
	Since Legendre transform reverses order, we have
	\begin{equation}
		\phi-|t|M=(\phi^*+|t|M)^*\leq (\phi^*+t\zeta)^*\leq (\phi^*-|t|M)^*=\phi +|t|M.
	\end{equation}
	Using the above estimates, we have the existence of $C>0$, such that
	\begin{equation}
	\begin{aligned}
		\left|\frac{h_t(x)-h(x)}{t}\right|&= |x|^{q-n} \left|\frac{e^{-\phi_t(x)}-e^{-\phi(x)}}{t}\right|\\&\leq |x|^{q-n} e^{-\phi(x)} \left|\max\left\{\frac{e^{|t|M}-1}{t}, \frac{e^{-|t|M}-1}{t}\right\}\right|\\
		&\leq 2C |x|^{q-n} e^{-\phi(x)},
	\end{aligned}
	\end{equation}
	for sufficiently small $|t|$. Therefore, by dominated convergence theorem, we have
	\begin{equation}
	\begin{aligned}
				\left.\frac{d}{dt}\right|_{t=0}\widetilde{V}_q(f_t) &= \int_{\mathbb R^n} \left.\frac{\partial }{\partial t}\right|_{t=0} h_t(x)dx\\
				& =\int_{\mathbb R^n} \zeta(\nabla \phi(x))|x|^{q-n}f(x)dx.
	\end{aligned}
	\end{equation}
\end{proof}

Let $A>0$ be an arbitrary number and $\mu$ be a nonzero even finite Borel measure on ${\mathbb R^n}$. Consider the following optimization problem
\begin{equation}
\label{eq local 4}
	\inf \left\{\int_{\mathbb R^n} \phi d\mu: \widetilde{V}_q(e^{-\phi^*})\geq  A, \phi \geq 0, \phi \in \L^1(\mu), \phi \text{ is an even function} \right\}.
\end{equation}

The following lemma shows that the Euler-Lagrange equation of the above constraint optimization problem implies the existence of solution to \eqref{eq local 2}.

\begin{lemma}
\label{lemma 13}
	Let $q>0$ and $\mu$ be a non-zero even finite Borel measure on ${\mathbb R^n}$ that is not concentrated in any proper subspaces. If an even function $\phi_0\in \cvx({\mathbb R^n})$ is such that $\phi_0\in \L^1(\mu)$, $\phi_0(o)>0$,
	\begin{equation}
		\widetilde{V}_q(e^{-\phi_0^*})= A,
	\end{equation}
	and
	\begin{equation}
		\int_{\mathbb R^n} \phi_0d\mu=\inf \left\{\int_{\mathbb R^n} \phi d\mu: \widetilde{V}_q(e^{-\phi^*})\geq A, \phi \geq 0, \phi \in \L^1(\mu),\phi \text{ is an even function} \right\},
	\end{equation}
	then
	\begin{equation}
		\mu = \widetilde{C}^{e}_{q}(f_0;\cdot),
	\end{equation}
	where $$f_0=\frac{|\mu|}{\widetilde{V}_q(e^{-\phi_0^*})}e^{-\phi_0^*}.$$
Moreover, $f_0\in \L^1$.
\end{lemma}
\begin{proof}
	Let $\zeta\in C_c({\mathbb R^n})$ be an even function. Set
	\begin{equation}
		\phi_t(x) = \phi_0(x)+t\zeta(x)
	\end{equation}
	and
	\begin{equation}
		\lambda(t) =\widetilde{V}_q(e^{-\phi_t^*}).
	\end{equation}
	The fact that $\lambda(t)$ is finite for small $|t|$ comes from Proposition \ref{prop 51}, the fact that $\phi_t\in \L^1(\mu)$, and that $\mu$ is a finite measure not concentrated in any proper subspaces.
	Set
	\begin{equation}
		\widetilde{\phi}_t(x) = \phi_t(x)-\ln \lambda(t)+\ln A.
	\end{equation}
	It is simple to notice that
	\begin{equation}
		\widetilde{V}_q(e^{-\widetilde{\phi}_t^*})=\widetilde{V}_q(e^{-\phi_t^*})\frac{A}{\lambda(t)}= A.
	\end{equation}
	
	Since $\zeta\in C_c({\mathbb R^n})$, there exists $M>0$ such that $|\zeta|<M$. This implies that
	\begin{equation}
		\lambda(t) \leq \widetilde{V}_q(e^{-\phi_0^*}) e^{|t|M} = Ae^{|t|M}.
	\end{equation}
	Thus, the choice of $\widetilde{\phi}_t$ implies that when $|t|$ is sufficiently small, we have
	\begin{equation}
		\widetilde{\phi}_t \geq \phi_0(o) -|t|M> 0.
	\end{equation}
	Such $\widetilde{\phi}_t$ belongs to the domain of the optimization problem. Since $\phi_0$ is the minimizer and that $\widetilde{\phi}_0 = \phi_0$, by Lemma \ref{lemma 51},
	\begin{equation}
	\begin{aligned}
		0 &= \left.\frac{d}{dt}\right|_{t=0} \int_{\mathbb R^n} \widetilde{\phi}_td\mu\\
			&= \int_{\mathbb R^n} \zeta(x)d\mu(x) - \frac{|\mu|}{A} \lambda'(0)\\
			& = \int_{\mathbb R^n} \zeta(x)d\mu(x) - \int_{\mathbb R^n} \zeta(x)d\widetilde{C}^{e}_{q}(f_0;x),
	\end{aligned}
	\end{equation}
	where $f_0=e^{-\phi_0^*+\ln |\mu|-\ln(A)}$. Since $\zeta$ is arbitrary in $C_c({\mathbb R^n})$, this implies that
	\begin{equation}
		\mu =  \widetilde{C}^{e}_{q}(f_0; \cdot).
	\end{equation}
	
	To see that $f_0\in \L^1$, note that since $\mu$ is not concentrated in any proper subspaces and $\phi_0\in \L^1(\mu)$, we conclude that $\phi_0$ is finite in a neighborhood of the origin. The fact that $f_0\in \L^1$ now follows from the Proposition \ref{prop 51} with $q=n$.
\end{proof}

Note that although the requirement $\phi\geq 0$ in the domain of the optimization problem is a closed condition, in order to make the Euler Lagrange equation equal to $0$, we have to establish that the minimizer actually satisfies a stronger condition ($ \phi_0>0)$. This ensures that the critical point is an interior point. This will be done in the next subsection.

\subsection{Existence of minimizer} This section is dedicated to showing the existence of a minimizer to \eqref{eq local 4} under the assumption that $\mu$ is an even measure.

It turns out that the $C^0$ estimates here are closely related to the estimates regarding the dual quermassintegrals of convex bodies. This is perhaps not surprising, given that in the case of characteristic function, the $(q-n)$-th moment of a log-concave function reduces to the $q$-th dual quermassintegral of a convex body. The following lemma reveals the simple fact that if the images of the orthogonal projections of a set of convex bodies onto a $k$-dimensional subspace, where $k=1,\dots, n-1$, are uniformly bounded, then their $q$-th dual quermassintegrals are uniformly bounded when $q<k$.

\begin{lemma}
\label{lemma 2}
	Let $k=1,\dots, n-1$, and $0<q<k$. For each $R>0$, there exists $C>0$,  such that for all $K\in \mathcal{K}^n_o$ satisfying
	\begin{equation}
	\label{eq 8181}
		P_\xi K \subset B(R)\cap \xi, \quad\text{ for some $k$-dim subspace }\xi\subset {\mathbb R^n},
	\end{equation}
we have
	\begin{equation}
		\widetilde{V}_q(K)<C.
	\end{equation}
Here, we use $P_\xi K$ to denote the image of the orthogonal projection of $K$ onto $\xi$.
\end{lemma}
\begin{proof}
	By \eqref{eq 8181}, we have
	\begin{equation}
		K\subset B(\sqrt{2}R)\cup [P_{\xi}K \times(\xi^{\perp}\setminus B(R))]\subset B(\sqrt{2}R)\cup [(B(R)\cap \xi) \times(\xi^{\perp}\setminus B(R))]
	\end{equation}
\mycomment{
	Indeed, if $(y,z)\in K\subset \xi\times \xi^\perp$, then, it must be the case that $y\in P_\xi K$.
	
	Case 1: if $|z|>R$, then $(y,z)\in P_{\xi}K \times(\xi^{\perp}\setminus B(R))$
	
	Case 2: otherwise $|z|\leq R$. Therefore, $|y|^2+|z|^2\leq |y|^2+R^2\leq R^2+R^2=2R^2$. Hence, $(y,z)\in B(\sqrt{2}R).$
	
	}
	Hence,
	\begin{equation}
	\begin{aligned}
		\widetilde{V}_q(K) &\leq \int_{\xi^\perp\setminus B(R)}\int_{B(R)\cap \xi} |(y,z)|^{q-n}d\mathcal{H}^k(y) d\mathcal{H}^{n-k}(z)+ \int_{B(\sqrt{2}R)}|x|^{q-n}dx\\
		&\leq \int_{\xi^\perp\setminus B(R)}|z|^{q-n}d\mathcal{H}^{n-k}(z)\int_{B(R)\cap \xi} d\mathcal{H}^k(y) + \frac{C}{q}(\sqrt{2}R)^q\\
		&= CR^k \int_{R}^{\infty} \rho^{q-n}\rho^{n-k-1}d\rho + \frac{C}{q}(\sqrt{2}R)^q\\
		&= C \frac{1}{k-q}R^{q} + \frac{C}{q}(\sqrt{2}R)^q.
	\end{aligned}
	\end{equation}
\end{proof}

A quick application of Lemma \ref{lemma 2} gives the following Blaschke-Santal\'{o} type estimates for normalized dual quermassintegrals. This is a special case of Theorem 3.1 in \cite{MR4259871}.
\begin{lemma}
\label{lemma 3}
	Let $K$ be an origin-symmetric convex body in ${\mathbb R^n}$. If $q\in (0,1)$ and $p>0$, then there exists $C>0$ independent of $K$ such that
	\begin{equation}
	\label{eq local 32}
		\overline{V}_{q}(K^*) \overline{V}_{p}(K)<C.
	\end{equation}
\end{lemma}
\begin{proof}
Let $v_0\in  {S^{n-1}}$ be such that $$h_{K^*}(v_0) = \min_{v\in  {S^{n-1}}} h_{K^*}(v).$$
Note that the functional $\overline{V}_{q}(K^*) \overline{V}_{p}(K)$ is invariant under rescaling of $K$. Therefore, by rescaling, we may assume $h_{K^*}(v_0)=1$.
This, by the choice of $v_0$, implies
\begin{equation}
	B\subset K^*,
\end{equation}
and consequently,
\begin{equation}
\label{eq 8182}
	K\subset B.
\end{equation}
Moreover,
\begin{equation}
	P_{\xi} K^*\subset B\cap \xi,
\end{equation}
where $\xi = \Span \{v_0\}$. Equation \eqref{eq local 32} follows from  Lemma \ref{lemma 2} and \eqref{eq 8182}.



\end{proof}

By integrating (in a certain way) over level sets of a log concave function, Lemma \ref{lemma 3} readily implies the following Blaschke-Santal\'{o} type inequality for log-concave functions.

\begin{lemma}
\label{lemma 4}
	Let $\phi\in \cvx({\mathbb R^n})$ be an even function with $\phi(o)=0$. Assume the effective domain of $\phi$ contains the origin as an interior point and $\lim_{|x|\rightarrow \infty}\phi(x)=\infty$. Suppose $q>0$ and $0<p<1$. There exists $C>0$, independent of $\phi$, such that
	\begin{equation}
		\left(\int_{\mathbb R^n} |x|^{q-n} e^{-\phi^*(x)}dx \right)\left(\int_0^\infty e^{-t}\overline{V}_{p}([\phi\leq t])^{q}dt\right)<C.
	\end{equation}
\end{lemma}
\begin{proof}
Note the fact that the effective domain of $\phi$ contains the origin as an interior point, that $\phi$ is even and convex, implies that $\phi$ is finite in a neighborhood of the origin. By Proposition \ref{prop 51}, this implies that
\begin{equation}
	\int_{\mathbb R^n} |x|^{q-n} e^{-\phi^*(x)}dx<\infty.
\end{equation}

	By definition of Legendre-Fenchel transform, we have
	\begin{equation}
		\phi^*(x) = \sup \{x\cdot y - \phi(y)\}\geq -\phi(o)=0.
	\end{equation}
	
	By the definition of Legendre-Fenchel transform, we have for any $s,t\geq 0$,
	\begin{equation}
		[\phi^*\leq s]\subset (s+t) [\phi\leq t]^*.
	\end{equation}
	Note that since $\phi$ is an even convex function whose effective domain contains the origin in its interior and $\phi(o)=0$, we conclude that for each $t>0$, the sublevel set $[\phi\leq t]$  contains the origin in the interior. On the other hand, since $\lim_{|x|\rightarrow \infty}\phi(x)=\infty$, the level set $[\phi\leq t]$ is bounded. Lemma \ref{lemma 3} now implies the existence of $C>0$ such that
\begin{equation}
	\overline{V}_{q}([\phi^*\leq s])\leq (s+t) \overline{V}_{q}([\phi\leq t]^*)\leq C(s+t) \overline{V}_{p}([\phi\leq t])^{-1}.
\end{equation}
By definition of $\overline{V}_{q}$, we have
\begin{equation}
\label{eq local 5}
	\int_{[\phi^*\leq s]} |x|^{q-n}dx \leq C (s+t)^{q} \overline{V}_{p}([\phi\leq t])^{-q}.
\end{equation}

Set
\begin{equation}
	F(s) = \begin{cases}
		e^{-s}\int_{[\phi^*\leq s]} |x|^{q-n}dx, &\text{ if } s\geq 0\\
		0, &\text{ otherwise,}
	\end{cases}
\end{equation}
and
\begin{equation}
	G(t) = \begin{cases}
		e^{-t}\overline{V}_{p}([\phi\leq t])^{q}, &\text{ if } t\geq 0\\
		0, &\text{ otherwise,}
	\end{cases}
\end{equation}
and
\begin{equation}
	H(x) = \begin{cases}
		\sqrt{C}e^{-x}(2x)^{\frac{q}{2}}, &\text{ if } x\geq 0,\\
		0, &\text{ otherwise.}
	\end{cases}
\end{equation}

Then, \eqref{eq local 5} implies that for any $s,t\geq 0$,
\begin{equation}
	H\left(\frac{1}{2}s +\frac{1}{2}t\right)= \sqrt{C}e^{-\frac{s+t}{2}}\left(s+t\right)^{\frac{q}{2}}\geq F\left(s\right)^{\frac{1}{2}}G\left(t\right)^{\frac{1}{2}}.
\end{equation}
It is simple to check that the above inequality is also true when $s$ or $t$ is negative, in which case the right-hand-side of the inequality is $0$.

Therefore, by Pr\'ekopa-Leindler inequality, we have
\begin{equation}
\label{eq local 6}
	\left(\int_0^\infty e^{-s}\int_{[\phi^*\leq s]} |x|^{q-n}dx ds\right) \left(\int_0^\infty e^{-t}\overline{V}_{p}([\phi\leq t])^{q}dt\right) \leq C \left(\int_{0}^\infty e^{-x} x^{\frac{q}{2}}dx\right)^2<C.
\end{equation}
(Recall that $C$ might change from line to line---in this case, even in the same line.)

The fact that $\phi^*\geq 0$ and layer-cake representation now imply
\begin{equation}
	\int_{\mathbb R^n} |x|^{q-n} e^{-\phi^*(x)}dx= \int_{0}^\infty \int_{[e^{-\phi^*}\geq t]} |x|^{q-n}dxdt = \int_{0}^\infty e^{-s}\int_{[\phi^*\leq s]} |x|^{q-n} dxds.
\end{equation}
This, when combined with \eqref{eq local 6}, gives the desired estimate.
\end{proof}

The above lemma immediately implies the following comparison.
\begin{lemma}
\label{lemma 11}
	Let $\phi\in \cvx({\mathbb R^n})$ be an even function with $\phi(o)=0$ and $\mu$ be a nonzero even finite Borel measure not concentrated in any proper subspaces. Suppose $q>0$ and $\phi\in \L^1(\mu)$. There exist $C_1>0$ and $C_2<0$, independent of $\phi$ such that
	\begin{equation}
	\label{eq local 6}
		\int_{\mathbb R^n} \phi d\mu \geq C_1 \widetilde{V}_q^{\frac{1}{q}}(e^{-\phi^*}) +C_2.
	\end{equation}
\end{lemma}

\begin{proof}
If
\begin{equation*}
	\widetilde{V}_q(e^{-\phi^*})=0,
\end{equation*}
or equivalently that $\phi^*$ is almost everywhere $+\infty$, there is nothing to prove. Therefore, we may assume that $\phi^*$ is finite in a neighborhood of the origin and by Proposition \ref{prop 51}, we have that
\begin{equation*}
	\int_{\mathbb R^n} e^{-\phi(x)}dx<\infty,
\end{equation*}
or equivalently, $\phi\rightarrow \infty$ as $|x|\rightarrow \infty$.

	We first note that since $\mu$ is not concentrated in any proper subspaces, there exists $C>0$ such that \begin{equation}
		\int_{\mathbb R^n} |x\cdot u|d\mu(x)>C,
	\end{equation}
	for every $u\in  {S^{n-1}}$.
	
	Set
	\begin{equation}
		K  = \{x\in {\mathbb R^n}: \phi(x)\leq 1\}.
	\end{equation}
	Since $\phi\in \L^1(\mu)$ is even and $\mu$ is not concentrated in any proper subspace, we have that $\phi$ is finite in a neighborhood of $o$. Therefore, by the fact that $\phi(o)=0$, we have that $K$ is a symmetric closed convex set that contains the origin in its interior.
	
	Let $r_K$ be such that
	\begin{equation}
		r_K = \sup\{r>0: rB\subset K\}.
	\end{equation}
	The facts that $\phi\rightarrow \infty$ when $|x|\rightarrow \infty$ imply that $K\neq {\mathbb R^n}$. Therefore $0<r_K<\infty$. Note that since $K$ is closed, there exists $u_0\in  {S^{n-1}}$ such that $r_K u_0\in \partial K$. This in turn implies that $h_K(u_0)=r_K$.
		
	We fix now an arbitrary $x\in {\mathbb R^n}$ with $|x\cdot u_0|>2r_K$. We consider $x' = \frac{2r_K}{|x\cdot u_0|}x$. Note that by choice of $x$, we have $|x'\cdot u_0|>r_K$. This implies that $x'\notin K$ and therefore $\phi(x')>1$. By convexity of $\phi$, we now have
	\begin{equation}
		1<\phi(x') = \phi\left(\left(1-\frac{2r_K}{|x\cdot u_0|}\right)o+\frac{2r_K}{|x\cdot u_0|}x\right)\leq  \frac{2r_K}{|x\cdot u_0|} \phi(x).
	\end{equation}
	Hence, for every $x\in {\mathbb R^n}$, we have
	\begin{equation}
		\phi(x)+1 \geq \frac{1}{2r_K} |x\cdot u_0|.
	\end{equation}
	Integrating with respect to $\mu$, we have
	\begin{equation}
	\label{eq local 7}
		\int_{\mathbb R^n} \phi d\mu \geq \frac{1}{2r_K}\int_{\mathbb R^n} |x\cdot u_0| d\mu(x)-|\mu|\geq \frac{C}{2r_K}-|\mu|.
	\end{equation}
	
	We now estimate the integral on the right-hand-side of \eqref{eq local 6}. By Lemma \ref{lemma 4}, there exists $0<p<1$ and $C>0$ such that
	\begin{equation}
	\label{eq local 8}
		\widetilde{V}_q(e^{-\phi^*})< C \left(\int_0^\infty e^{-t}\overline{V}_{p}([\phi\leq t])^{q}dt\right)^{-1}.
	\end{equation}
	
	Note that by the convexity of $\phi$ and that $\phi(o)=0$, we have
	\begin{equation}
		[\phi\leq t]\supset t K,
	\end{equation}
	for $t<1$. Therefore, we have
	\begin{equation}
	\label{eq local 9}
	\begin{aligned}
		\int_0^\infty e^{-t}\overline{V}_{p}([\phi\leq t])^{q}dt&\geq \int_{0}^1 e^{-t} \overline{V}_{p}(tK)^{q}dt \\
		& = \overline{V}_{p}(K)^{q} \int_{0}^1 e^{-t}t^{q}dt\\
		&\geq C\overline{V}_{p}(r_KB)^{q}\\
		&= C r_K^{q}.
	\end{aligned}
	\end{equation}
	
	The desired inequality now follows from \eqref{eq local 7}, \eqref{eq local 8}, and \eqref{eq local 9}.
	\end{proof}
	
	
In \cite{MR3341966}, Cordero-Erausquin and Klartag demonstrated that, if $\phi_k$ is a sequence of nonnegative functions with uniform $\L^1(\mu)$ bound and $\phi_k$ attains minimum at the origin, then one may construct a pointwise convergent subsequence (to a convex function) such that the $\L^1(\mu)$ norm of $\phi_k$ is lower semi-continuous and $\widetilde{V}_n(e^{-\phi_k^*})$ is upper semi-continuous with respect to such convergence. Upon a careful inspection of the proof, it is not hard to see that such ``selection theorem'' holds true for any $q>0$. We state the following generalized version without providing a proof.

\begin{lemma}[\cite{MR3341966}]\label{lemma selection}
	Let $q>0$ and $\mu$ be a non-zero even finite Borel measure on ${\mathbb R^n}$. Assume $\mu$ is not entirely concentrated in any proper subspaces. If $\phi_k\in \cvx({\mathbb R^n})$ is non-negative, even and
	\begin{equation}
		\sup_k \int_{\mathbb R^n} \phi_k d\mu<\infty,
	\end{equation}
	then, there exists a subsequence $\phi_{k_j}$ that converges pointwise to a non-negative, even, convex function $\phi\in \L^1(\mu)$ such that
	\begin{equation}
	\label{eq 62}
		\int_{\mathbb R^n} \phi d\mu\leq \liminf_{k\rightarrow\infty} \int_{\mathbb R^n} \phi_kd\mu, \quad \text{and}\quad \widetilde{V}_q(e^{-\phi^*})\geq \limsup_{k\rightarrow\infty} \widetilde{V}_q(e^{-\phi_k^*}).
	\end{equation}
\end{lemma}
	
We are now ready to show the existence of minimizer to \eqref{eq local 4}.
\begin{lemma}
\label{lemma 12}
	Let $q>0$ and $\mu$ be a non-zero even finite Borel measure on ${\mathbb R^n}$. Suppose $\mu$ is not entirely concentrated in any proper subspaces and $\int_{\mathbb R^n} |x|d\mu(x)<\infty$. For sufficiently large $A>0$, there exists an even function $\phi_0 \in \cvx({\mathbb R^n})$ such that $\phi_0\in \L^1(\mu)$, $\phi_0(o)>0$,
	\begin{equation}
		\widetilde{V}_q(e^{-\phi_0^*})= A,
	\end{equation}
	and
	\begin{equation}
	\label{eq local 14}
		\int_{\mathbb R^n} \phi_0d\mu=\inf \left\{\int_{\mathbb R^n} \phi d\mu: \widetilde{V}_q(e^{-\phi^*})\geq  A, \phi \geq 0, \phi \in \L^1(\mu),\phi \text{ is an even function} \right\}.
	\end{equation}
\end{lemma}

\begin{proof}
Let $c_n>0$ be such that
$$\int_{B(c_n)} |x|^{q-n}dx =1.$$
Set
\begin{equation}
	C = c_n\int_{\mathbb R^n} |x|d\mu>0.
\end{equation}
By the condition on $\mu$, it is simple to see that $C$ is finite.

We choose a fixed $A>0$ such that
	\begin{equation}
	\label{eq 8191}
		|\mu|\ln A+ C<C_1A^{\frac{1}{q}}+C_2,
	\end{equation}
	where $C_1, C_2$ are from Lemma \ref{lemma 11} and depend only on $\mu$ and $q$. It is simple to see that such an $A$ exists and in fact, all sufficiently large $A>0$ satisfies \eqref{eq 8191}.
	
	Let $\phi_k$ be a minimizing sequence; that is, we have $\phi_k\geq 0$, $\phi_k\in \L^1(\mu)$, $\phi_k$ is even,
	\begin{equation}
		\widetilde{V}_q(e^{-\phi_k^*}) \geq A,
	\end{equation}
	and
	\begin{equation}
		\lim_{l\rightarrow \infty} \int_{\mathbb R^n} \phi_k d\mu =\inf \left\{\int_{\mathbb R^n} \phi d\mu: \widetilde{V}_q(e^{-\phi^*})\geq  A, \phi \geq 0, \phi \in \L^1(\mu),\phi \text{ is an even function} \right\}.
	\end{equation}
	
	Notice that
	\begin{equation}
		\phi_k^{**}\leq \phi_k,
	\end{equation}
	and since $\phi_k\geq 0$, we have $\phi_k^{**}\geq 0$. Moreover, note that $(\phi_k^{**})^*=\phi_k^*$. Therefore, we may without loss of generality, assume that $\phi_k \in \cvx({\mathbb R^n})$.
	
	We claim now that
\begin{equation}
\label{eq 61}
	\sup_{k} \int_{\mathbb R^n} \phi_k d\mu<\infty.
\end{equation}
To see this, set
\begin{equation}
	\Gamma(x) = \ln A + c_n|x|,
\end{equation}
It is simple to compute that
	\begin{equation}
		\Gamma^*(x) = 1^\infty_{B(c_n)}(x)-\ln A,
	\end{equation}
	and that
	\begin{equation}
		\widetilde{V}_q(e^{-\Gamma^*})=A,
	\end{equation}
	according to the choice of $c_n$. Note that $\Gamma(x)$ is positive and even. Moreover,
	\begin{equation}
	\label{eq local 15}
		\int_{\mathbb R^n} \Gamma(x)d\mu = |\mu|\ln A+ C,
	\end{equation}
	Since $\phi_k$ is a minimizing sequence, we conclude \eqref{eq 61}.
	
	By Lemma \ref{lemma selection}, there exists a subsequence $\phi_{k_j}$ that converges pointwise to a non-negative even convex function $\phi_0$ such that \eqref{eq 62} holds true. In particular, this suggests that $\phi_0$ is minimizer to \eqref{eq local 14}.  By possibly replacing $\phi_0$ by $\phi_0^{**}$, we may assume that $\phi_0\in \cvx({\mathbb R^n})$.
	
	It remains to show that $\phi_0(o)>0$ and
	\begin{equation}
	\label{eq 63}
		\widetilde{V}_q(e^{-\phi_0^*})= A.
	\end{equation}
	
	To see the former, we argue by contradiction. Assume $\phi_0(o)=0$. Therefore, we may conclude by using Lemma \ref{lemma 11} that
	\begin{equation}
		|\mu|\ln A+ C\geq \int_{\mathbb R^n} \phi_0 d\mu \geq C_1 \widetilde{V}_q^{\frac{1}{q}}(e^{-\phi_0^*}) +C_2\geq C_1A^{\frac{1}{q}}+C_2.
	\end{equation}
	This is a contradiction to \eqref{eq 8191}.
	
	To show \eqref{eq 63}, if it was not the case, we set
	\begin{equation}
		\widetilde{\phi}_0 = \phi_0-\varepsilon.
	\end{equation}
	Note that for sufficiently small $\varepsilon>0$, we have $\widetilde{\phi}_0>0$ thanks to $\phi_0(o)>0$. Moreover,
	\begin{equation}
		\widetilde{V}_q(e^{-\widetilde{\phi}_0^*})= e^{-\varepsilon}\widetilde{V}_q(e^{-\phi_0^*}) >A,
	\end{equation}
	for sufficiently small $\varepsilon>0$. However, it is trivial to see
	\begin{equation}
		\int_{\mathbb R^n} \widetilde{\phi}_0d\mu<\int_{\mathbb R^n} {\phi}_0d\mu,
	\end{equation}
	which contradicts with the minimality of $\phi_0$.
\end{proof}

Lemmas \ref{lemma 13} and \ref{lemma 12} now immediately solves the Minkowski problem \eqref{eq local 2}.
\begin{theorem}
\label{thm 500}
	Let $q>0$ and $\mu$ be a non-zero even finite Borel measure on ${\mathbb R^n}$. Suppose $\mu$ is not entirely concentrated in any proper subspaces and $\int_{\mathbb R^n} |x|d\mu(x)<\infty$. There exists an even $f_0\in \Lc({\mathbb R^n})$ with nonzero finite $\L^1$ norm such that
	\begin{equation}
		\mu = \widetilde{C}^{e}_{q}(f_0;\cdot).
	\end{equation}
\end{theorem}

To complete this section, we show that the assumption that
\begin{equation}
	\int_{\mathbb R^n} |x|d\mu(x)<\infty
\end{equation}
is necessary in Theorem \ref{thm 500}. We require the following basic lemma about log-concave function.
\begin{lemma}
\label{lemma log concave derivative control}
	Suppose $f: \mathbb{R}\rightarrow [0,\infty)$ be a log-concave function. If
	\begin{equation}
	\label{eq 2001}
		\lim_{t\rightarrow \pm \infty} f(t)=0,
	\end{equation}
	then, for each $t_0\geq 0$, we have
	\begin{equation}
		\int_{|t|\geq t_0} |f'(t)|dt\leq 4\sup_{|t|\geq t_0} f(t).
	\end{equation}
\end{lemma}
\begin{proof}
	Since $f$ is log-concave, it is locally Lipschitz in the interior of the interval in which it is positive. Moreover, it is unimodal. Therefore, by \eqref{eq 2001} and fundamental theorem of Calculus, we have
	\begin{equation}
		\int_{t\geq t_0} |f'(t)|dt\leq 2 \sup_{t\geq t_0} f(t),
	\end{equation}
	and
	\begin{equation}
		\int_{t\leq -t_0} |f'(t)|dt\leq 2 \sup_{t\leq -t_0} f(t).
	\end{equation}
	Combining the above two inequality gives us the desired result.
\end{proof}
\begin{theorem}
\label{thm necessity}
	Let $q>0$ and $f\in \Lc({\mathbb R^n})$ be with nonzero finite $\L^1$ norm. If $K_f$, the support of $f$, contains the origin as an interior point, then
	\begin{equation}
	\label{eq 1002}
		\int_{{\mathbb R^n}} |x|  d\widetilde{C}^{e}_{q}(f;x) = \int_{{\mathbb R^n}} |\nabla f(x)| \cdot |x|^{q-n}dx<\infty.
	\end{equation}
	In particular, \eqref{eq 1002} is valid for even $f\in \Lc({\mathbb R^n})$ with nonzero finite $\L^1$ norm.
\end{theorem}
\begin{proof}
We first consider the case where $q\in (0,n]$.

Since $K_f$ contains the origin as an interior point, the function $f$ is Lipschitz in $B(r_0)$ for some $r_0>0$. Denote the Lipschitz constant of $f$ inside $B(r_0)$ by $M$. This implies that inside $B(r_0)$, we have $|\nabla f|\leq M$ almost everywhere. Therefore,
\begin{equation}
	\int_{B(r_0)} |\nabla f(x)| \cdot |x|^{q-n}dx\leq M\int_{B(r_0)}|x|^{q-n}dx<\infty.
\end{equation}

Therefore, to show \eqref{eq 1002}, we only need to show
\begin{equation}
		\int_{B(r_0)^c} |\nabla f(x)| \cdot |x|^{q-n}dx<\infty.
\end{equation}
In particular, it is sufficient to show
	\begin{equation}
		\int_{B(r_0)^c} \left|\frac{\partial f}{\partial x_n}(x)\right|\cdot |x|^{q-n}dx<\infty.
	\end{equation}
	We write $x=(y,t)$. Note that since $f\in \L^1$, then $f\rightarrow 0$ as $|x|\rightarrow \infty$. Therefore, since $0<q\leq n$, by Lemma \ref{lemma log concave derivative control},
	\begin{equation}
	\label{eq 1000}
	\begin{aligned}
		&\int_{B(r_0)^c} \left|\frac{\partial f}{\partial t}(x)\right| \cdot |x|^{q-n}dx\\
		\leq& \left(\int_{B(r_0/2)^c\cap \mathbb{R}^{n-1}}\int_{-\infty}^{\infty}+\int_{B(r_0/2)\cap \mathbb{R}^{n-1}}\int_{|t|\geq \frac{\sqrt{3}}{2}r_0}\right) \left|\frac{\partial f}{\partial t}(y,t)\right||(y,t)|^{q-n}dtdy\\
		\leq &4\int_{B(r_0/2)^c\cap \mathbb{R}^{n-1}} \sup_{t\in \mathbb{R}} f(y,t)\cdot |y|^{q-n}dy+ \left(\frac{\sqrt{3}}{2}r_0\right)^{q-n} \int_{B(r_0/2)\cap \mathbb{R}^{n-1}}\int_{|t|\geq \frac{\sqrt{3}}{2}r_0}\left|\frac{\partial f}{\partial t}(y,t)\right|dtdx\\
		\leq & 4\int_{B(r_0/2)^c\cap \mathbb{R}^{n-1}} \sup_{t\in \mathbb{R}} f(y,t)\cdot |y|^{q-n}dy + 4\left(\frac{\sqrt{3}}{2}r_0\right)^{q-n}\int_{B(r_0/2)\cap \mathbb{R}^{n-1}}\sup_{t\in \mathbb{R}}f(y,t)dy\\
		\leq &4\int_{B(r_0/2)^c\cap \mathbb{R}^{n-1}} \sup_{t\in \mathbb{R}} f(y,t)\cdot |y|^{q-n}dy + C,
	\end{aligned}
	\end{equation}
	for some positive constant $C$ depending on $r_0$ and $\sup f$.
	Since $f\in \L^1$, if we write $f=e^{-\phi}$,  we have
	\begin{equation}
	\label{eq 2000}
		\liminf_{|x|\rightarrow \infty}\frac{\phi(x)}{|x|}>0.
	\end{equation}
	In particular, this implies the existence of $c_0>0$ and $M>0$ such that for all $|(y,t)|>M$, we have
	\begin{equation}
	\label{eq 1001}
		\phi(y,t)>c_0 |(y,t)|.
	\end{equation}
	The desired result follows from combining \eqref{eq 1000} and \eqref{eq 1001}.
	
	Let us now consider the case $q>n$. 	
	
	We first note that
	\begin{equation}
	\label{eq 1004}
		\int_{B(M^2)} |\nabla f(x)|\cdot |x|^{q-n}dx\leq M^{2(q-n)} \int_{B(M^2)}|\nabla f(x)|dx\leq M^{2(q-n)} \int_{{\mathbb R^n}} |\nabla f(x)|dx<\infty,
	\end{equation}
	where the last inequality follows from the previous established case $q=n$. Therefore, we only need to show
	\begin{equation}
	\label{eq 2002}
		\int_{B(M^2)^c} |\nabla f(x)|\cdot |x|^{q-n}dx<\infty.
	\end{equation}
	Note that, by \eqref{eq 1001}
	\begin{equation}
	\label{eq 1005}
		\begin{aligned}
			&\int_{B(M^2)^c} \left|\frac{\partial f}{\partial x_n} (x)\right|\cdot |x|^{q-n}dx \\=&\sum_{k=M^2}^{\infty} \int_{B(k+1)\setminus B(k)}  \left|\frac{\partial f}{\partial x_n} (x)\right|\cdot |x|^{q-n}dx\\
			\leq & \sum_{k=M^2}^{\infty}(k+1)^{q-n} \int_{B(k+1)\setminus B(k)} \left|\frac{\partial f}{\partial x_n} (x)\right|dx\\
			\leq& \sum_{k=M^2}^{\infty}(k+1)^{q-n}\left(\int_{[B(k+1)\setminus B(k-1)]\cap \mathbb{R}^{n-1}}\int_{-\infty}^{\infty}+\int_{B(k-1)\cap \mathbb{R}^{n-1}}\int_{|t|\geq \sqrt{2k}}\right) \left|\frac{\partial f}{\partial t}(y,t)\right|dtdy\\
					\end{aligned}
	\end{equation}
	
	For the first term, by \eqref{eq 1001}, we have
	\begin{equation}
	\label{eq 2003}
		\begin{aligned}
			&\sum_{k=M^2}^{\infty}(k+1)^{q-n}\int_{[B(k+1)\setminus B(k-1)]\cap \mathbb{R}^{n-1}}\int_{-\infty}^{\infty} \left|\frac{\partial f}{\partial t}(y,t)\right|dtdy\\
			\leq & 4\sum_{k=M^2}^{\infty}(k+1)^{q-n}\int_{[B(k+1)\setminus B(k-1)]\cap \mathbb{R}^{n-1}}\sup_{t\in \mathbb{R}}f(y,t)dy\\
			\leq & 4\sum_{k=M^2}^\infty (k+1)^{q-n}\int_{[B(k+1)\setminus B(k-1)]\cap \mathbb{R}^{n-1}} e^{-c_0 |y|}dy\\
			\leq & 4C\sum_{k=M^2}^\infty(k+1)^{q-n}e^{-c_0(k-1)}(k+1)^{n-1}<\infty.
		\end{aligned}
	\end{equation}
	
	For the second term, by using Lemma \ref{lemma log concave derivative control} again, we have
	\begin{equation}
		\int_{|t|\geq \sqrt{2k}}\left|\frac{\partial f}{\partial t}(y,t)\right|dt\leq 4\sup_{|t|\geq \sqrt{2k}} f(y,t),
	\end{equation}
	and consequently, by \eqref{eq 1001},
	\begin{equation}
	\label{eq 2004}
	\begin{aligned}
		&\sum_{k=M^2}^{\infty}(k+1)^{q-n}\int_{B(k-1)\cap \mathbb{R}^{n-1}}\int_{|t|\geq \sqrt{2k}} \left|\frac{\partial f}{\partial t}(y,t)\right|dtdy\\
		\leq&  4\sum_{k=M^2}^{\infty}(k+1)^{q-n}\int_{B(k-1)\cap \mathbb{R}^{n-1}}\sup_{|t|\geq \sqrt{2k}} f(y,t)dy\\
		\leq & 4\sum_{k=M^2}^{\infty}(k+1)^{q-n}\int_{B(k-1)\cap \mathbb{R}^{n-1}}e^{-c_0\sqrt{2k}}dy\\
		=& 4C\sum_{k=M^2}^{\infty}(k+1)^{q-n} (k-1)^{n-1} e^{-c_0\sqrt{2k}}<\infty.
	\end{aligned}	
	\end{equation}
	Equation \eqref{eq 2002} now follows from \eqref{eq 1005}, \eqref{eq 2003}, and \eqref{eq 2004}.
\end{proof}

\mycomment
\subsection{Essential continuity of solution}

We remark here that following an argument in \cite{MR3341966}, the solution we obtained in Theorem \ref{thm 500} (or, equivalently, in Lemma \ref{lemma 12}) is essentially continuous. We require the following inequality that follows immediately from the fact that we obtained our solution through solving a minimization problem.
\begin{lemma}
\label{lemma inequality}
	Under the assumption of Lemma \ref{lemma 12}, if $|\mu|=1$, we further have
	\begin{equation}
	\label{eq 1006}
		\log \int_{{\mathbb R^n}}|x|^{q-n}e^{-\phi_0^*}dx-\log \int_{{\mathbb R^n}} |x|^{q-n}e^{-\phi^*}dx\geq \int_{{\mathbb R^n}}(\phi_0-\phi)d\mu,
	\end{equation}
	as long as $\phi\in \cvx({\mathbb R^n})$ is an even function and $\phi\in \L^1(\mu)$.
\end{lemma}
\begin{proof}
	Note that if we further have $\phi\geq 0$ and
	\begin{equation}
		\int_{{\mathbb R^n}}|x|^{q-n}e^{-\phi^*(x)}dx= A,
	\end{equation}
	then \eqref{eq 1006} follows since $\phi_0$ is the minimizer to the minimization problem \eqref{eq local 14}.
	
	For a generic $\phi$, we consider
	\begin{equation}
		\phi_c = \phi+c.
	\end{equation}
	Note that $\phi_c^* = \phi^*-c$. Therefore,
	\begin{equation}
	\label{eq 1007}
		\int_{{\mathbb R^n}}|x|^{q-n}e^{-\phi_c^*(x)}dx = \int_{{\mathbb R^n}}|x|^{q-n}e^{-\phi^*(x)}dx\cdot e^c,
	\end{equation}
	and
	\begin{equation}
	\label{eq 1008}
		\int_{\mathbb R^n} \phi_c d\mu= \int_{\mathbb R^n} \phi d\mu+ c.
	\end{equation}
	By choosing $c$ to be sufficiently big, since $\phi\in \cvx({\mathbb R^n})$, we have
	\begin{equation}
		\phi_c\geq 0 \qquad \text{and }\qquad \int_{{\mathbb R^n}}|x|^{q-n}e^{-\phi_c^*(x)}dx\geq A
	\end{equation}
	and therefore \eqref{eq 1006} holds with $\phi= \phi_c$. Now, because of \eqref{eq 1007} and \eqref{eq 1008}, we have the desired result.
\end{proof}


The following lemma follows from the same arguments used in \cite[Proposition 10]{MR3341966}. We therefore only provide a short sketch, pointing out only the necessary changes.
\begin{lemma}
\label{lemma 501}
	If $|\mu|=1$, the $\phi_0^*$ we obtained from Lemma 5.8 is essentially continuous.
\end{lemma}
\begin{proof}

	There are only two differences between what we want to prove here and in \cite[Proposition 10]{MR3341966}.
	
	First, here, we restrict our attention to the symmetric setting and as a consequence, in order to apply \eqref{eq 1006} to get the desired comparison, we have to pick an origin symmetric $\phi$. The modification is simple: pick
	\begin{equation}
		\phi_\varepsilon(y) = \max \{\phi_0(y), \phi_0(y)+\varepsilon |\theta\cdot y|-1\}
	\end{equation}
	in \cite[(25)]{MR3341966}. Although we will not have (30) there as a consequence of this change, we can still get the estimates in the unnumbered equation below (30), which is all that is needed.
	
	The other difference is that here we consider
	\begin{equation}
		\int_{{\mathbb R^n} } |x|^{q-n}e^{-\phi^*(x)}dx
	\end{equation}
	instead of the special case $q=n$. However, notice that the proof works locally for an arbitrary fixed $x_0\in S$. Since $\phi_0$ is even, $S$ must be positively away from the origin. Therefore, in all the estimates that follow, the term $|x|^{q-n}$ actually has uniform positive upper and lower bound.
	
	Aside from these two minor changes, the rest of the proof follows verbatim with the help of Theorem \ref{thm necessity}.
\end{proof}
\mycomment{
\begin{proof}
	We provide a detailed proof here, for self-reference.
	
	Denote
	\begin{equation}
		K = \{\phi_0^*<\infty\}, \qquad\text{and}\qquad S= \partial K.
	\end{equation}
	The set $K$ is convex and has non-empty interior---because otherwise
	\begin{equation}
		\int_{{\mathbb R^n}}|x|^{q-n} e^{-\phi_0^*(x)}dx=0.
	\end{equation}
	At interior points of $K$, the convex function $\phi_0^*$ is continuous. We only need to inspect discontinuous points on $S$. Since $\phi_0^*$ is lower semi-continuous, if $\phi_0^*(x_0)=\infty$ at some $x_0\in S$, then $x_0$ must be a continuous point. Therefore, the only discontinuous points on $S$ are
	\begin{equation}
		A= \{x\in S: \phi_0^*(x)<\infty\}.
	\end{equation}
	
	We need to show $\mathcal{H}^{n-1}(A)=0$.
	
	Note that it is sufficient to show that for every $x\in A$, there exists $U(x)\subset S$, an open neighborhood in $S$ of $x$, such that $\mathcal{H}^{n-1}(U(x)\cap A)=0$. (If this can be done, then in particular, we can choose neighborhoods with rational center and radius and there are only countably many of them.)
	
	We fix $x_0\in A$. The boundary $S=\partial K$ is locally the graph of a convex function in an appropriate coordinate system. This means that there exists an open neighborhood $U_0\subset S$ of $x_0$ in $S$ and $\theta\in  {S^{n-1}}$ and $c_0>0$ such that
	\begin{equation}
	\label{eq 1010}
		x+t\theta\notin \overline{K}, \quad \forall x\in U_0, \forall t>0,
	\end{equation}
	and
	\begin{equation}
	\label{eq 1011}
		|P_{\theta^\perp} x-P_{\theta^\perp}y|\geq c_0|x-y|, \quad \forall x, y\in U_0.
	\end{equation}
	The latter relation is because of the fact that convex functions are locally Lipschitz. We may also choose $U_0$ to be small enough so that there exists $c_1>0$ such that
	\begin{equation}
	\label{eq 1013}
		\frac{1}{c_1}<|x|<c_1, \quad \forall x\in U_0,
	\end{equation}
	owing to the fact that $K$ must contain the origin as an interior point.
	
	Let
	\begin{equation}
		A^k  = \{x\in U_0\cap A: \phi_0^*(x)\leq k\}.
	\end{equation}
	Note that the sets $A^k$ are increasing in $k$ and $\cup A_k = A\cap U_0$. Therefore, we only need to show that for every $k\geq 1$, we have $\mathcal{H}^{n-1}(A^{k})=0$.  For this purpose, in the following, we fix an arbitrary $k\geq 1$.
	
	For sufficiently small $\varepsilon>0$, we consider
	\begin{equation}
		\phi_\varepsilon(y) = \max \{\phi_0(y), \phi_0(y)+\varepsilon |y\cdot \theta|-1\}.
	\end{equation}
	Note that $\phi_\varepsilon$ is even, convex and lower semi-continuous. Since $\mu$ is finite and has finite first moment, we conclude that $\phi_\varepsilon\in \L^1(\mu)$. Therefore, according to Lemma \ref{lemma inequality}, we have
	\begin{equation}
	\label{eq 1014}
		\log \int_{{\mathbb R^n}} |x|^{q-n}e^{-\phi_\varepsilon^*}dx-\log \int_{{\mathbb R^n}}|x|^{q-n}e^{-\phi_0^*}dx\leq \int_{{\mathbb R^n}}(\phi_\varepsilon-\phi_0)d\mu.
	\end{equation}
	We now estimate both sides of the above inequality as $\varepsilon\rightarrow 0$.


Consider $B_\varepsilon=\{y\in {\mathbb R^n} : |y|\leq \frac{1}{\varepsilon}\}$. We have
\begin{equation}
	\varepsilon|y\cdot \theta|-1\leq 0, \quad \forall y\in B_\varepsilon.
\end{equation}
Therefore,
\begin{equation}
\begin{aligned}
	\int_{\mathbb R^n} \phi_\varepsilon - \phi_0 d\mu &=\int_{\mathbb R^n} \max \{0, \varepsilon|y\cdot \theta|-1\}d\mu(y)\\
	&= \int_{{\mathbb R^n} \setminus B_\varepsilon}\max \{0, \varepsilon|y\cdot \theta|-1\}d\mu(y)\\
	&\leq \varepsilon \int_{{\mathbb R^n} \setminus B_\varepsilon} |y|d\mu(y).
\end{aligned}
\end{equation}
Since $\mu$ has finite first moment, we have that
\begin{equation}
	\lim_{\varepsilon\rightarrow 0}\int_{{\mathbb R^n} \setminus B_\varepsilon} |y|d\mu(y)=0.
\end{equation}
Therefore, we have
\begin{equation}
\label{eq 1015}
	\int_{\mathbb R^n} \phi_\varepsilon - \phi_0 d\mu\leq o(\varepsilon),
\end{equation}
as $\varepsilon\rightarrow 0$.

By definition of $\phi_\varepsilon$ and the fact that Legendre-Fenchel transformation reverses order, we have
\begin{equation}
	\phi_\varepsilon^*\leq \phi_0^*.
\end{equation}
Also,
\begin{equation}
	\phi_\varepsilon^*(x)\leq (\phi_0+\varepsilon|\langle\cdot, \theta\rangle|)^*(x)+1 = \sup_{y\in {\mathbb R^n}}(x\cdot y -\phi_0(y)-\varepsilon|y\cdot \theta|)+1.
\end{equation}
For every $x\in A^{k}$ and $t\in[0,\varepsilon]$, we have
\begin{equation}
\label{eq 1012}
	\phi_\varepsilon^*(x+t\theta) = \sup_{y\in {\mathbb R^n}} (x\cdot y - \phi_0(y)+t\theta\cdot y-\varepsilon|y\cdot \theta|)+1\leq \sup_{y\in {\mathbb R^n} }(x\cdot y-\phi_0(y))+1=\phi_0^*(x)+1\leq k+1,
\end{equation}
and
\begin{equation}
	\phi_\varepsilon(x)\leq k.
\end{equation}

Define
\begin{equation}
	A_\varepsilon^k = \{x+t\theta: x\in A^k , t\in (0, \varepsilon]\}.
\end{equation}
By \eqref{eq 1010} and \eqref{eq 1011}, we have
\begin{equation}
	\mathcal{H}^n(A_\varepsilon^k)= \varepsilon \mathcal{H}^{n-1}(P_{\theta^\perp} A^{k})\geq \varepsilon c_0^{n-1}\mathcal{H}^{n-1}(A^k).
\end{equation}
By \eqref{eq 1010} and \eqref{eq 1012}, we have
\begin{equation}
	A_\varepsilon^k\cap K = \emptyset, \quad \text{and} \quad \sup_{z\in A_\varepsilon^k} \phi_\varepsilon^*(z)\leq k+1.
\end{equation}

By \eqref{eq 1013}, by requiring $\varepsilon>0$ to be small enough, we get
\begin{equation}
	\frac{1}{2c_1}<|x|^{q-n}<2c_1, \quad \forall x\in A_\varepsilon^k.
\end{equation}

Therefore, we have
\begin{equation}
	\begin{aligned}
		\int_{\mathbb R^n} |x|^{q-n}e^{-\phi_\varepsilon^*(x)}dx &= \left(\int_{{\mathbb R^n}\setminus K}+\int_{K}\right)|x|^{q-n}e^{-\phi_\varepsilon^*(x)}dx\\
		&\geq \int_{A_\varepsilon^k}|x|^{q-n}e^{-\phi_\varepsilon^*(x)}dx+\int_K |x|^{q-n}e^{-\phi_0^*(x)}dx\\
		&\geq \frac{1}{2c_1}e^{-(k+1)}\varepsilon c_0^{n-1}\mathcal{H}^{n-1}(A^k) + \int_{\mathbb R^n} |x|^{q-n}e^{-\phi_0^*(x)}dx.
	\end{aligned}
\end{equation}
Consequently,
\begin{equation}
	\log \int_{{\mathbb R^n}} |x|^{q-n}e^{-\phi_\varepsilon^*}dx\geq \log \int_{{\mathbb R^n}} |x|^{q-n}e^{-\phi_0^*(x)}dx+\log (1+ \varepsilon\cdot C)\geq \log \int_{{\mathbb R^n}} |x|^{q-n}e^{-\phi_0^*(x)}dx+\varepsilon\cdot C/2,
\end{equation}
for sufficiently small $\varepsilon>0$, where
\begin{equation}
	C = \frac{1}{2c_1}e^{-(k+1)} c_0^{n-1}\mathcal{H}^{n-1}(A^k)/\int_{\mathbb R^n} |x|^{q-n}e^{-\phi_0^*(x)}dx.
\end{equation}
Hence,
\begin{equation}
\label{eq 1016}
	\log \int_{{\mathbb R^n}} |x|^{q-n}e^{-\phi_\varepsilon^*}dx-\log \int_{{\mathbb R^n}} |x|^{q-n}e^{-\phi_0^*(x)}dx\geq \varepsilon \cdot C/2.
\end{equation}
Combining \eqref{eq 1014}, \eqref{eq 1015}, and \eqref{eq 1016}, we have $C=0$ or, equivalently, $\mathcal{H}^{n-1}(A^k)=0$.
\end{proof}

\begin{theorem}
	Suppose $f\in \Lc({\mathbb R^n})$ is even and has finite $\L^1$ norm. If $f$ is essentially continuous, then the finite Borel measure $\widetilde{C}^{e}_{q}(f;\cdot)$ is not concentrated entirely in any proper subspaces.
\end{theorem}
\begin{proof}
	We argue by contradiction. Without loss of generality, suppose $\widetilde{C}^{e}_{q}(f; \cdot)$ is concentrated entirely in a proper subspace $e_n^\perp$. As a consequence
	\begin{equation}
		0=\int_{{\mathbb R^n} }|x\cdot e_n| d\widetilde{C}^{e}_{q}(f;x) = \int_{{\mathbb R^n}} \left|\frac{\partial}{\partial x_n}f(x)\right||x|^{q-n}dx.
	\end{equation}
	Hence, for almost all $x\in {\mathbb R^n}$, we have
	\begin{equation}
		\frac{\partial}{\partial x_n}f(x)=0.
	\end{equation}
	Argue as in Lemma 6 in Klartag, we have $f(y,t)$ is constant in $t$ and therefore contradicts with the fact that $f$ has nonzero finite $\L^1$ norm.
\end{proof}}

\begin{theorem}
	Let $q>0$ and $\mu$ be a non-zero even finite Borel measure on ${\mathbb R^n}$. There exists an even, essentially continuous $f_0\in \Lc({\mathbb R^n})$ with nonzero finite $\L^1$ norm such that
	\begin{equation}
		\mu = \widetilde{C}^{e}_{q}(f_0;\cdot),
	\end{equation}
	if and only if  $\mu$ is not entirely concentrated in any proper subspaces and $\int_{\mathbb R^n} |x|d\mu(x)<\infty$.
\end{theorem}
\begin{proof}
By homogeneity, we may assume $|\mu|=1$.

	The sufficiency follows from Theorem \ref{thm 500}, and Lemma \ref{lemma 501}. The necessity follows from Theorem \ref{thm necessity} and \cite[Lemma 6]{MR3341966}.
\end{proof}}


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