© Wuhan University 2025

0 Introduction

In 1897, Minkowski posed the Minkowski problem^[1]. Minkowski^[1-2] resolved its discrete 3D case. For arbitrary measures, Aleksandrov^[3], Fenchel and Jessen^[4] established complete solutions via variational methods, requiring the convex body's centroid to lie at the origin.

Lutwak^[5] established the $L_p$-Brunn-Minkowski theory, formulating the $L_p$ Minkowski problem^[6-9]. Further generalization emerges through Orlicz-Brunn-Minkowski theory^[10-14], extending the $L_p$ framework^[15].

Recent studies have extended Minkowski problems to Gaussian spaces. Huang et al^[16] introduced the Gaussian surface area measure $S_{{\gamma }_n,K}$, establishing the Gaussian-Minkowski problem with existence and uniqueness in classical cases. Feng et al^[17] proposed $L_p$ Gaussian surface area measures and corresponding Minkowski problems, proving existence in normalized settings.

Let ${\mathbb{R}}^n$ denote the $n$-dimensional Euclidean space. Let $n$ denote a vector in ${\mathbb{R}}^n$. The $s$-Gaussian probability measure ${\gamma }_n^{ s}$ for $E\subset {\mathbb{R}}^n$ and $s\in (\mathrm{0},\mathrm{\infty })$ is outlined in Ref. [18] as follows:

${\gamma }_n^{ s}(E)=\frac{\mathrm{1}}{C_{s,n}}{\int }_E{\left(\mathrm{1}-\frac{s|\mathit{x}{|}^{\mathrm{2}}}{\mathrm{2}}\right)}_{+}^{\frac{\mathrm{1}}{s}} \mathrm{d}\mathit{x}$(1)

where $C_{s,n}:=\frac{n{\omega }_n}{\mathrm{2}}{\sqrt[]{\frac{\mathrm{2}}{s}}}^n\beta (\frac{\mathrm{1}}{s}+\mathrm{1} , \frac{n}{\mathrm{2}})$, and ${\omega }_n$ is the volume of the unit ball. When $s\to {\mathrm{0}}^{+}$, ${\gamma }_n^{ s}$ is the classical Gaussian probability measure

${\gamma }_n(E)=\frac{\mathrm{1}}{{(\sqrt[]{\mathrm{2}\mathrm{\pi }})}^n}{\int }_E{\mathrm{e}}^{- \frac{ {\left|\mathit{x}\right|}^{\mathrm{2}}}{\mathrm{2}}} \mathrm{d}\mathit{x}$

The generalized Gaussian probability space^[19-20] features the generalized Gaussian density $g_{\alpha ,q}$. Ref. [21] introduced a generalized Gaussian surface area measure. Moreover, Liu et al^[21] proposed the generalized Gaussian-Minkowski problem and established the existence and uniqueness of solutions in the smooth case.

Definition 1   Let $p>\mathrm{1}$, $s>\mathrm{0}$ and $K\in {\mathcal{K}}_o^n$. The $L_{p,s}$-Gaussian surface area measure of $K$, denoted by $S_{{\gamma }_n^{ s},p,K}$, is a Borel measure on $S^{n-\mathrm{1}}$ given by

$S_{{\gamma }_n^{ s},p,K}(\eta )=\frac{\mathrm{1}}{p{C}_{s,n}}{\int }_{{\mathit{v}}_K^{-\mathrm{1}}(\eta )}{\left(\mathit{x}\cdot {\mathit{v}}_K(\mathit{x})\right)}^{\mathrm{1}-p}{\left(\mathrm{1}-\frac{s|\mathit{x}{|}^{\mathrm{2}}}{\mathrm{2}}\right)}_{+}^{\frac{\mathrm{1}}{s}}\mathrm{d}{\mathrm{\mathscr{H}}}^{n-\mathrm{1}}(\mathit{x})$(2)

for any Borel measurable $\eta \subset S^{n-\mathrm{1}}$. When $s\to {\mathrm{0}}^{+}$, then the $L_{p,s}$-Gaussian surface area measure is the $L_p$-Gaussian surface area measure.

The $L_{p,s}$-Gaussian-Minkowski Problem: If $p>\mathrm{1}$, given a finite Borel measure $\mu$ on the $S^{n-\mathrm{1}}$, what is the sufficient and necessary condition on the measure $\mu$ so that there exists a convex body $K\in {\mathcal{K}}_o^n$ and $\alpha \in (\mathrm{0},\frac{\mathrm{1}}{n})$ such that $\mu =\frac{S_{{\gamma }_n^{ s},p,K}}{{\gamma }_n^{ s}{(K)}^{\mathrm{1}-\alpha }}$.

The following theorem present a solution to the $L_{p,s}$-Gaussian-Minkowski problem for the case of even measures.

Theorem 1   Suppose $\mu$ is a non-zero finite even Borel measure not concentrated in any closed hemisphere and $\alpha \in (\mathrm{0},\frac{\mathrm{1}}{n})$. Then there exists an o-symmetric convex body $K\in {\mathcal{K}}_e^n$ for $s\in (\mathrm{0},\mathrm{\infty })$ such that $\mu =\frac{S_{{\gamma }_{ n}^{ s},p, K_s}}{{\gamma }_{ n}^{ s}(K_s{)}^{\mathrm{1}-\alpha }}$.

1 Preliminaries

We introduce some notations and, for ease of reference, summarize some basic properties of convex bodies. For comprehensive treatments of convex body theory, we refer readers to the texts by Gardner^[22], Gruber^[23], and Schneider^[24], etc.

Let $\mathit{x}\in {\mathbb{R}}^n$, the norm of a vector $\mathit{x}$ is given by $|\mathit{x}|=\sqrt[]{\mathit{x}\cdot \mathit{x}}$, $\overline{\mathit{x}}$ denotes $\frac{\mathit{x}}{\left|\mathit{x}\right|}$. The unit sphere in ${\mathbb{R}}^n$ is denoted by $S^{n-\mathrm{1}}$. Let $C(S^{n-\mathrm{1}})$ represent the set of continuous functions on $S^{n-\mathrm{1}}$. Furthermore, $C_e^{+}(S^{n-\mathrm{1}})$ denotes the set of even, positive, continuous functions on $S^{n-\mathrm{1}}$. For a finite measure $\mu$ on $S^{n-\mathrm{1}}$, the total mass of $\mu$ is denoted by $|\mu | =\mu (S^{n-\mathrm{1}})$.

1.1 Convex Bodies

The set of convex bodies (compact convex subsets) in ${\mathbb{R}}^n$ is denoted by ${\mathcal{K}}^n$, the set of convex bodies containing the origin in their interiors is denoted by ${\mathcal{K}}_o^n$, and ${\mathcal{K}}_e^n\subset {\mathcal{K}}_o^n$ denotes the set of origin-symmetric convex bodies. The unit ball centered at the origin in ${\mathbb{R}}^n$ is denoted by $B^n$, and its volume by ${\omega }_n$. $\partial K$ denotes the boundary of $K\in {\mathcal{K}}^n$.

The support function, $h_K:S^{n-\mathrm{1}}\to \mathbb{R}$, of a nonempty compact convex set $K$, is the continuous function on the unit sphere $S^{n-\mathrm{1}}$, defined by $h_K(\mathit{u})=\mathrm{m}\mathrm{a}\mathrm{x}\{\mathit{u}\cdot \mathit{x}:\mathit{x}\in K\}.$ The radial function, ${\rho }_K:S^{n-\mathrm{1}}\to \mathbb{R}$ for $K$ is the continuous function, defined by ${\rho }_K(\mathit{x})=\mathrm{m}\mathrm{a}\mathrm{x}\{\lambda :\lambda \mathit{x}\in K\}.$

For $K\in {\mathcal{K}}_o^n$, the polar body $K^{ \mathrm{*}}$ of $K$ is defined by $K^{ \mathrm{*}}=\{\mathit{x}\in {\mathbb{R}}^n:\mathit{y}\in K,\mathit{x}\cdot \mathit{y}\le \mathrm{1}\}.$

Let $h\in C^{+}(S^{n-\mathrm{1}})$, the Wulff shape $[h]$ associated with $h$ be defined as $[h]=\{\mathit{x}\in {\mathbb{R}}^n:\mathit{x}\cdot \mathit{v}\le h(\mathit{v}), \mathit{v}\in S^{n-\mathrm{1}}\}.$ The collection of nonempty compact convex sets can be viewed as a metric space with the Hausdroff metric, where the Hausdorff distance between $K$ and $L$ is defined by

$d(K,L)=\mathrm{m}\mathrm{i}\mathrm{n}\{t\ge \mathrm{0}:K\subset L+t{B}^n,L\subset K+t{B}^n\}.$

1.2 The Radial Gauss Map of a Convex Body

For $K\in {\mathcal{K}}_o^n$ and each $\mathit{v}\in S^{n-\mathrm{1}}$, the hyperplane

$H_K(\mathit{v})=\{\mathit{x}\in {\mathbb{R}}^n:\mathit{x}\cdot \mathit{v}=h_K(\mathit{v})\}$

is called the supporting hyperplane to $K$ with unit normal $\mathit{v}$. For $\sigma \subset \partial K$, the Gaussian image of $\sigma$ is defined by

${\mathit{v}}_K(\sigma )=\{\mathit{v}\in S^{n-\mathrm{1}}: \mathit{x}\in H_K(\mathit{v})\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{o}\mathrm{m}\mathrm{e} \mathit{x}\in \sigma \}\subset S^{n-\mathrm{1}}.$

For $\eta \subset S^{n-\mathrm{1}}$, the reverse Gaussian image of $\eta$ is defined by ${\mathit{v}}_K^{-\mathrm{1}}(\eta )=\{\mathit{x}\in \partial K:\mathit{x}\in H_K(\mathit{v})\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{o}\mathrm{m}\mathrm{e}\mathit{v}\in \eta \}\subset \partial K.$

Huang et al^[25] gave the definitions of radial Gaussian image. For $K\in {\mathcal{K}}_o^n$ and $\eta \subset S^{n-\mathrm{1}}$, the radial Gaussian image ${\mathit{\alpha }}_K(\mathit{u})$ is the set of outer unit normals of $K$ from the boundary points ${\rho }_K(\mathit{u}) \mathit{u}$, for some $\mathit{u}\in \eta$; that is,

${\mathit{\alpha }}_K(\eta )=\underset{\mathit{u}\in \eta }{\cup }\{\mathit{v}\in S^{n-\mathrm{1}}:{\rho }_K(\mathit{u}) \mathit{u}\cdot \mathit{v}=h_{ K}(\mathit{v})\}.$

Denote ${\omega }_K\subset S^{n-\mathrm{1}}$ as the set of $\mathit{u}\in S^{n-\mathrm{1}}$ such that ${\mathit{\alpha }}_K(\mathit{u})$ contains more than one point; that is, the point ${\rho }_K(\mathit{u}) \mathit{u}\in \partial K$ has more than one outer unit normal. We now define the radial Gauss map ${\alpha }_K:S^{n-\mathrm{1}}\backslash {\omega }_K\to S^{n-\mathrm{1}},$ satisfying ${\mathit{\alpha }}_K\left(\{\mathit{u}\}\right)=\left\{{\alpha }_K\left(\mathit{u}\right)\right\}$.

2 Variational Formulas for the ${\mathit{L}}_{\mathit{p},\mathit{s}}$-Gaussian Surface Area Measure

In this section, we derive the variational formula for the $L_{p,s}$-Gaussian surface area measure.

Lemma 1   (Ref.[16], Lemma 3.2) Let $K\in {\mathcal{K}}_o^n$ and $g\in C(S^{n-\mathrm{1}})$. Suppose $\delta >\mathrm{0}$ is sufficiently small so that for each $t\in (-\delta ,\delta )$, there is $h_{ t}=h_K+t g>\mathrm{0}$. Then,

$\underset{t\to \mathrm{0}}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{{\rho }_{[h_{ t}]}(\mathit{u})-{\rho }_K(\mathit{u})}{t}=\frac{g({\alpha }_K(\mathit{u}))}{h_K({\alpha }_K(\mathit{u}))}{\rho }_K(\mathit{u})$

for almost all $\mathit{u}\in S^{n-\mathrm{1}}$ with respect to spherical Lebesgue measure. Moreover, there exists a constant $M>\mathrm{0}$, such that

$\left|{\rho }_{[h_{ t}]}(\mathit{u})-{\rho }_K(\mathit{u})\right|<M\left|t\right|$(3)

for all $\mathit{u}\in S^{n-\mathrm{1}}$ and $t\in (-\delta ,\delta )$.

Theorem 2   Let $K\in {\mathcal{K}}_o^n$, $s\in (\mathrm{0},\mathrm{\infty })$, $g\in C(S^{n-\mathrm{1}})$. Then,

$\underset{t\to \mathrm{0}}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{{\gamma }_n^{ s}([{\left(h_{ K}^{ p}+t g\right)}^{\frac{\mathrm{1}}{p}}])-{\gamma }_n^{ s}(K)}{t}={\int }_{S^{n-\mathrm{1}}}g \mathrm{d}{S}_{{\gamma }_n^{ s},p,K}.$

Proof   Let $h_{ t}={\left(h_{ K}^{ p}+t g\right)}^{\frac{\mathrm{1}}{p}}$, then

$\begin{array}{l}{h}_{ t}={\left(h_{ K}^{ p}+t g\right)}^{\frac{\mathrm{1}}{p}}=h_{ K}+\frac{\mathrm{1}}{p}{h}_{ K}^{ p(\frac{\mathrm{1}}{p}-\mathrm{1})}g t+o(t)\\ =h_{ K}+\frac{\mathrm{1}}{p}{h}_{ K}^{ \mathrm{1}-p}g t+o(t),\end{array}$(4)

where $o(t)$ is the same order infinitesimal of $t$. It is evident that $[h_{ t}]\in {\mathcal{K}}_o^n$ when $t$ is sufficiently small. By the definition of the $s$-Gaussian probability measure in (1) and employing polar coordinates, we obtain the following expression:

$\begin{array}{l}{\gamma }_{ n}^{ s}\left(\left[h_{ t}\right]\right)=\frac{\mathrm{1}}{C_{s,n}}{\int }_{[h_{ t}]}{\left(\mathrm{1}-\frac{s|\mathit{x}{|}^{\mathrm{2}}}{\mathrm{2}}\right)}_{+}^{\frac{\mathrm{1}}{s}}d\mathit{x}\\ =\frac{\mathrm{1}}{C_{s,n}}{\int }_{S^{n-\mathrm{1}}}{\int }_{\mathrm{0}}^{{\rho }_{ [h_{ t}]}(\mathit{u})}{\left(\mathrm{1}-\frac{s|r\mathit{u}{|}^{\mathrm{2}}}{\mathrm{2}}\right)}_{+}^{\frac{\mathrm{1}}{s}}{r}^{n-\mathrm{1}} \mathrm{d}r \mathrm{d}\mathit{u}.\end{array}$(5)

Denote $F(t)={\int }_{\mathrm{0}}^t{\left(\mathrm{1}-\frac{s|r\mathit{u}{|}^{\mathrm{2}}}{\mathrm{2}}\right)}_{+}^{\frac{\mathrm{1}}{s}} r^{n-\mathrm{1}} \mathrm{d}r$. By the mean value theorem and (3), we have

$\begin{array}{l}\left|F({\rho }_{ [h_{ t}]}(\mathit{u}))-F({\rho }_K(\mathit{u}))\right|=\left|F^{\text{'}}(\theta )\right| \left|\rho { }_{[h_{ t}]}(\mathit{u})-{\rho }_K(\mathit{u})\right|\\ \le M\left|F^{\text{'}}(\theta )\right| \left|t\right|,\end{array}$

where $\theta$ is between ${\rho }_{ [h_{ t}]}(\mathit{u})$ and ${\rho }_K(\mathit{u})$. Therefore, by definition of $F$, we have $|F^{\text{'}}(\theta )|$ is bounded from above by some constant $M_{\mathrm{1}}$. Furthermore, there exists $M_{\mathrm{2}}=M_{\mathrm{1}}M>\mathrm{0}$ such that

$|F({\rho }_{ [h_{ t}]}(\mathit{u}))-F(\rho (\mathit{u}))|\le M_{\mathrm{2}} t$

Together with (5), the dominated convergence theorem, Lemma 1, (4) and (2), we attain

$\begin{array}{l}\underset{t\to \mathrm{0}}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{{\gamma }_{ n}^{ s}([{\left(h_{ K}^{ p}+t g\right)}^{\frac{\mathrm{1}}{p}}])-{\gamma }_{ n}^{ s}(K)}{t}\\ =\underset{t\to \mathrm{0}}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{\mathrm{1}}{C_{s,n}}{\int }_{S^{n-\mathrm{1}}}\frac{\mathrm{1}}{t}{{\int }_{{\rho }_K(\mathit{u})}^{{\rho }_{ [h_{ t}]}(\mathit{u})}{\left(\mathrm{1}-\frac{s|r\mathit{u}{|}^{\mathrm{2}}}{\mathrm{2}}\right)}_{+}^{\frac{\mathrm{1}}{s}}r}^{n-\mathrm{1}} \mathrm{d}r \mathrm{d}\mathit{u}\\ =\frac{\mathrm{1}}{C_{s,n}}{\int }_{S^{n-\mathrm{1}}}\frac{\mathrm{1}}{p} h_K^{\mathrm{1}-p}(\mathit{u}) g({\alpha }_K(\mathit{u})){\left(\mathrm{1}-\frac{s{\rho }_K{(\mathit{u})}^{\mathrm{2}}}{\mathrm{2}}\right)}_{+}^{\frac{\mathrm{1}}{s}}\frac{{\rho }_K{(\mathit{u})}^n}{h_K({\alpha }_K(\mathit{u}))} \mathrm{d}\mathit{u}\\ =\frac{\mathrm{1}}{C_{s,n}}{\int }_{{\partial }^{\text{'}}K}\frac{\mathrm{1}}{p} (\mathit{x}\cdot {\mathit{v}}_K{(\mathit{x}))}^{\mathrm{1}-p} g\left({\mathit{v}}_K\left(\mathit{x}\right)\right){\left(\mathrm{1}-\frac{s{\left|\mathit{x}\right|}^{\mathrm{2}}}{\mathrm{2}}\right)}_{+}^{\frac{\mathrm{1}}{s}}\mathrm{d}{\mathrm{\mathscr{H}}}^{n-\mathrm{1}}(\mathit{x})\\ ={\int }_{S^{n-\mathrm{1}}}g(\mathit{u}) \mathrm{d}{S}_{{\gamma }_{ n}^{ s},p, K}(\mathit{u}),\end{array}$

which is the desired conclusion.

3 The ${\mathit{L}}_{\mathit{p},\mathit{s}}$-Gaussian-Minkowski Problem and Its Solution

In this section, we focus on the normalized version of the $L_{p,s}$-Gaussian-Minkowski problem for even measure when $s\in (\mathrm{0},\mathrm{\infty })$. For $\mathrm{0}<\alpha <\frac{\mathrm{1}}{n}$ and $p>\mathrm{1}$, we first transform the problem of the existence of solutions to the $L_{p,s}$-Gaussian-Minkowski problem into the following maximisation problem:

$\mathrm{s}\mathrm{u}\mathrm{p}\{F_s(f):f\in C_e^{+}(S^{n-\mathrm{1}})\}$(6)

where $F_s:C_e^{+}\left(S^{n-\mathrm{1}}\right)\to \mathbb{R}$ is given for $f\in C_e^{+}\left(S^{n-\mathrm{1}}\right)$ by

$F_s(f)=\frac{\mathrm{1}}{\alpha }{\gamma }_{ n}^{ s}{\left(\left[f\right]\right)}^{\alpha }-{\int }_{S^{n-\mathrm{1}}}{f}^p \mathrm{d}\mu$(7)

For $K\in {\mathcal{K}}_e^n$, denote $F_s(K)=F_s(h_K)$.

Lemma 2   Let $\mathrm{0}<\alpha <\frac{\mathrm{1}}{n}$ and $p>\mathrm{1}$. If an even function $f_s$ is a maximizer to the optimization problem (6), then $f_s$ must be the support function of an o-symmetric convex body; that is, there exists an o-symmetric convex body $K_s\in {\mathcal{K}}_e^n$ such that $f_s=h_{K_s}$. Moreover, $K_s$ satisfies

$\mu =\frac{S_{{\gamma }_{ n}^{ s},p, K_s}}{{\gamma }_{ n}^{ s}(K_s{)}^{\mathrm{1}-\alpha }}$(8)

Proof   Note that for each $f\in C_e^{+}(S^{n-\mathrm{1}})$, $h_{ [f]}\le f$ and the Wulff shape of $h_{ [f]}$ is the same as that of $f$ . So according to (7), we have $F_s(f)\le F_s(h_{ [f]})$.

Therefore, the maximizer of (6) $f_s$ must be the support function of some o-symmetric convex body $K_s\in {\mathcal{K}}_e^n$. We now use the variational formula to establish that $K_s$ satisfies (8). To this end, for each $g\in C_e(S^{n-\mathrm{1}})$, consider the family of Wulff shapes

$K_t=\left[{\left(h_{K_s}^p+tg\right)}^{\frac{\mathrm{1}}{p}}\right]\in {\mathcal{K}}_e^n,$

for $t$ small enough.

Since $f_s=h_{K_s}$ is a maximizer of (6) and the definition of $F_s$ given in equation (7) and by Theorem 2, it follows that

$\mathrm{0}=\frac{\mathrm{d}}{\mathrm{d}t}|{}_{t=\mathrm{0}}{F}_s(h_{K_t})={\gamma }_{ n}^{ s}(K_s{)}^{\alpha -\mathrm{1}}{\int }_{S^{n-\mathrm{1}}}g \mathrm{d}{S}_{{\gamma }_{ n}^{ s},p, K_s}-{\int }_{S^{n-\mathrm{1}}}g \mathrm{d}\mu .$

Note that the above equation holds for every $g\in C_e(S^{n-\mathrm{1}})$. Therefore, we conclude that $K_s$ satisfies (8).

Lemma 3   Let $\mathrm{0}<\alpha <\frac{\mathrm{1}}{n}$, $p>\mathrm{1}$ and $s\in (\mathrm{0},\mathrm{\infty })$. For sufficiently small $r>\mathrm{0}$, we have $F_s(r{B}^n)>\mathrm{0}$.

Proof   From the definition of $F_s$ given in equation (7), we have

$\begin{array}{l}{F}_s(r{B}^n)=\frac{\mathrm{1}}{\alpha }{\left(\frac{\mathrm{1}}{C_{s,n}}{\int }_{r{B}^n}{\left(\mathrm{1}-\frac{s|\mathit{x}{|}^{\mathrm{2}}}{\mathrm{2}}\right)}_{+}^{\frac{\mathrm{1}}{s}}d\mathit{x}\right)}^{\alpha }-{\int }_{S^{n-\mathrm{1}}}{r}^p \mathrm{d}\mu \\ =\frac{\mathrm{1}}{\alpha }{\left(\frac{n {\omega }_n}{C_{s,n}}{\int }_{\mathrm{0}}^r{\left(\mathrm{1}-\frac{s t^{\mathrm{2}}}{\mathrm{2}}\right)}_{+}^{\frac{\mathrm{1}}{s}} t^{n-\mathrm{1}} \mathrm{d}t\right)}^{\alpha }-r^p|\mu |\\ =r\left[\frac{\mathrm{1}}{\alpha }{\left(\frac{n {\omega }_n}{C_{s,n}}\right)}^{\alpha }{\left(\frac{{\int }_{\mathrm{0}}^r{\left(\mathrm{1}-\frac{s t^{\mathrm{2}}}{\mathrm{2}}\right)}_{+}^{\frac{\mathrm{1}}{s}} t^{n-\mathrm{1}} \mathrm{d}t}{r^{\frac{\mathrm{1}}{\alpha }}}\right)}^{\alpha }-r^{p-\mathrm{1}}|\mu |\right].\end{array}$(9)

It follows from L'Hôpital's rule^[26] that when $\mathrm{0}<\alpha <\frac{\mathrm{1}}{n}$, we have

$\begin{array}{l}\underset{r\to {\mathrm{0}}^{+}}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{{\int }_{\mathrm{0}}^r{\left(\mathrm{1}-\frac{s t^{\mathrm{2}}}{\mathrm{2}}\right)}_{+}^{\frac{\mathrm{1}}{s}} t^{n-\mathrm{1}} \mathrm{d}t}{r^{\frac{\mathrm{1}}{\alpha }}}=\underset{r\to {\mathrm{0}}^{+}}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{\alpha {\left(\mathrm{1}-\frac{s t^{\mathrm{2}}}{\mathrm{2}}\right)}_{+}^{\frac{\mathrm{1}}{s}} r^{n-\mathrm{1}}}{r^{{}^{\frac{\mathrm{1}}{\alpha }-\mathrm{1}}}}\\ =\alpha \underset{r\to {\mathrm{0}}^{+}}{\mathrm{l}\mathrm{i}\mathrm{m}}{r}^{{ }^{n-\frac{\mathrm{1}}{\alpha }}}=\mathrm{\infty }\end{array}$(10)

According to (9), (10) and $p>\mathrm{1}$, for $r$ is sufficiently small, $F_s(r{B}^n)>\mathrm{0}$.

Proof of Theorem 1   By Lemma 2, it suffices to show that a maximizer to the optimization problem (6) exists. We assume that $\{K_i{\}}_{i=\mathrm{1}}^{\mathrm{\infty }}$ is a sequence of o-symmetric convex bodies and

$\underset{i\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}{F}_s(K_i)=\mathrm{s}\mathrm{u}\mathrm{p}\{F_s(f):f\in C_e^{+}(S^{n-\mathrm{1}})\}\ge F_s(r{B}^n)>\mathrm{0}.$

Choose $r_i>\mathrm{0}$ and ${\mathit{u}}_i\in S^{n-\mathrm{1}}$ such that $r_i {\mathit{u}}_i\in K_i$ and $r_i=\underset{\mathit{u}\in S^{n-\mathrm{1}}}{\mathrm{m}\mathrm{a}\mathrm{x}}{\rho }_{K_i}(\mathit{u})$.

It is simple to notice that $K_i\subset r_i{B}^n$. We claim that $r_i$ is a bounded sequence from above with upper bound $M$. Otherwise, by taking a subsequence $r_{i_j}$, we may assume that $\underset{j\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}{r}_{i_j}=+\mathrm{\infty }$. Since $K_{i_j}\subset r_{i_j}{B}^n$ and (7), we have

$F_s(K_{i_j})\le \frac{\mathrm{1}}{\alpha }{\gamma }_{ n}^{ s}(r_{i_j}{B}^n{)}^{\alpha }-{\int }_{S^{n-\mathrm{1}}}{h}_{K_{i_j}}^p \mathrm{d}\mu .$

Since $r_{i_j}{\mathit{u}}_{i_j}\in K_{i_j}$, we have $h_{K_{i_j}}(\mathit{v})\ge r_{i_j} {\mathit{u}}_{i_j}\cdot \mathit{v}$. Therefore, we have

$F_s(K_{i_j})\le \frac{\mathrm{1}}{\alpha }{\gamma }_{ n}^{ s}(r_{i_j}{B}^n{)}^{\alpha }-r_{i_j}^p{\int }_{\{\mathit{v}\in S^{n-\mathrm{1}}: \mathit{v}\cdot {\mathit{u}}_{i_j}\ge \mathrm{0}\}}|\mathit{v}\cdot {\mathit{u}}_{i_j}{|}^p\mathrm{d}\mu$(11)

By the fact that $\mu$ is even and not concentrated in any closed hemisphere, there exists a constant $c_{\mathrm{0}}>\mathrm{0}$ such that

${\int }_{\{\mathit{v}\in S^{n-\mathrm{1}}: \mathit{v}\cdot {\mathit{u}}_{i_j}\ge \mathrm{0}\}}|\mathit{v}\cdot {\mathit{u}}_{i_j}{|}^p\mathrm{d}\mu \ge c_{\mathrm{0}}$(12)

as $r_{i_j}\to +\mathrm{\infty }$, ${\gamma }_{ n}^{ s}(r_{i_j}{B}^n)\to {\gamma }_{ n}^{ s}({\mathbb{R}}^n)=\mathrm{1}$. Therefore, according to (11), (12),

$F_s(K_{i_j})\le \frac{\mathrm{1}}{\alpha }{\gamma }_{ n}^{ s}(r_{i_j}{B}^n{)}^{\alpha }-r_{i_j}^p c_{\mathrm{0}}^p<\mathrm{0}$

as $j\to \mathrm{\infty }$. The above equation is in contradiction to

$\underset{j\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}{F}_s(K_{i_j})=\mathrm{s}\mathrm{u}\mathrm{p}\{F_s(f):f\in C_e^{+}(S^{n-\mathrm{1}})\}>\mathrm{0}.$

This leads to a contradiction. Therefore, the sequence of convex bodies $\{K_i{\}}_{i=\mathrm{1}}^{\mathrm{\infty }}$ is uniformly bounded.

We use Blaschke selection theorem and assume (by taking a subsequence) that $\{K_i{\}}_{i=\mathrm{1}}^{\mathrm{\infty }}$ converges to some $K_{\mathrm{0}}\in {\mathcal{K}}_e^n$ in Hausdorff metric. Note that by the continuity of the s-Gaussian volume with respect to the Hausdorff metric, definition of $F_s$, and $F_s(r B^n)>\mathrm{0}$, we have

$\frac{\mathrm{1}}{\alpha }{\gamma }_{ n}^{ s}(K_{\mathrm{0}}{)}^{\alpha }\ge F_s(K_{\mathrm{0}})=\underset{i\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}{F}_s(K_i)>\mathrm{0}.$

Combining the inequality mentioned above, with the fact that $K_{\mathrm{0}}$ is o-symmetric, implies that $K_{\mathrm{0}}$ contains the origin in its interior. Therefore, the convex body $K_{\mathrm{0}}$ is a maximizer to the optimization (6).

References