The $L_{p}$-Gaussian Minkowski problem

The Brunn-Minkowski theory mainly studies the geometric functional of convex bodies and their differential. The differentiable functional will generate some intrinsic geometric measures. In Euclidean space $\mathbb{R}^{n}$, the most significant functional of convex body is volume denoted by $V$, and Aleksandrov [1] gave the variational formula

where $K+tL=\{x+ty:\text{$x\in K$ and $y\in L$}\}$, $h_{L}:S^{n-1}\rightarrow\mathbb{R}$ is the support function of convex body $L$. It naturally generates the surface area measure $S_{K}$ of convex body $K$. The classical Minkowski problem characterizes the surface area measure, it reads : given a finite Borel measure $\mu$ on $S^{n-1}$, what are the necessary and sufficient conditions on $\mu$ does there exist a convex body $K$ in $\mathbb{R}^{n}$ such that surface area measure $S_{K}(\cdot\,)=\mu(\cdot\,)$? If $K$ exists, is it unique? In the smooth case, it is equivalent to solve a certain fully nonlinear elliptic equation. The classical Minkowski problem was solved by Minkowski in [26] for polytope, and later solved by Aleksandrov [1], Fenchel and Jessen [16] in general.

In 1993, Lutwak developed the Brunn-Minkowski theory to $L_{p}$-Brunn-Minkowski theory. The $L_{p}$-Minkowski problem, characterizing $L_{p}$-surface area measure $S_{p}(K)={h_{K}}^{1-p}S_{K}$. Clearly, when $p=1$, the $L_{p}$-surface area measure is just the classical surface measure. Lutwak [39] testified that there exists a unique $o$-symmetry convex body such that $S_{p}(K)=\mu$ for $p>1$. The $L_{p}$-Minkowski problem has been studied extensively in [3, 4, 11, 27, 24, 33, 43, 48, 47, 49] and so on. The uniqueness of the Minkowski problem is related to Brunn-Minkowski inequality. For $p>1$, the $L_{p}$-Brunn-Minkowski inequality can be obtained easily. In case of $0<p<1$, $L_{p}$-Brunn-Minkowski inequality and its condition of the equality holds have not been improved much until [8]. While the logarithmic Minkowski problem $(p=0)$ and the centro-affine Minkowski problem $(p=-n)$ are two special cases; see, e.g., [5], and [12]. The regularity of the $L_{p}$-Minkowski problem, for example [12, 28, 40].

In [38], the dual Brunn-Minkowski theory was developed in the 1970s. The most significant dual curvature measure and its associated Minkowski problem in the dual Brunn-Minkowski theory were studied in [29]. The work of [29] play a key role in promoting the development of partial differential equation and dual Brunn-Minkowski theory as [9, 31, 6, 41, 10, 45]. Recently, a number of meaningful and interesting work emerged on the Orlicz-Minkowski problem, such as [19, 20, 25, 34, 35, 46, 52].

In [30], the authors posed the Minkowski problem in Gaussian probability space. Gaussian volume functional denoted by $\gamma_{n}$ plays a central role, which is given by

Using the variational formula from Huang et al. [29], the variational formula of $\gamma_{n}$ is obtained as follows:

where Gaussian surface area measure $S_{\gamma_{n},K}$ is given by

About the mainly result of Gaussian Minkowski problem showed in [30] Theorem 1.4, stated as: Let $\mu$ be a finite even Borel measure on $S^{n-1}$ that is not concentrated on any closed hemisphere and $|\mu|<\frac{1}{\sqrt{2\pi}}$. Then there exists a unique $o$-symmetry convex body with $\gamma_{n}(K)>\frac{1}{2}$ such that $S_{\gamma_{n},K}=\mu.$

As remarked in [30], Gaussian Minkowski problem is different from the Minkowski problem in Lebesgue measure space in these aspects. At first, the Gaussian surface area of all convex set in $\mathbb{R}^{n}$ is not more than $4{n^{\frac{1}{4}}}$ which is shown detailed in Ball [2] and [18]. That is to say, the allowable $\mu$ in Gaussian Minkowski problem can not have an arbitrarily big total mass. Secondly, Gaussian probability measure has neither translation invariance nor homogeneity, which makes our research more difficult. As a final remark, by the definition of Gaussian surface area, we can know that not only small convex bodies can have smaller Gaussian surface areas, but also large convex bodies. For instance, let $B_{r}$ be a centered ball with radius $r,$ the Gaussian surface area’s density of $B_{r}$ is $f_{r}=\frac{1}{(\sqrt{2\pi})^{n}}e^{\frac{-r^{2}}{2}}r^{n-1}$. One can observe that $\frac{1}{(\sqrt{2\pi})^{n}}e^{\frac{-r^{2}}{2}}r^{n-1}\rightarrow 0$ as $r\rightarrow 0$ or $r\rightarrow\infty.$

Naturally, we develop the $L_{p}$-Brunn-Minkowski theory in Gaussian probability space. The purpose of this paper is to continue the study in [30] and to construct $L_{p}$-Gaussian surface area measures. A $L_{p}$-variational formula similar to Lemma 3.2 in Lutwak [39] is gained as below:

where $h_{t}(v)=(h_{K}(v)^{p}+tf(v)^{p})^{\frac{1}{p}}$ is $L_{p}$-combination for $p\neq 0$, and it leads to define the $L_{p}$-Gaussian surface area measures as follows

The $L_{p}$-Gaussian Minkowski problem reads : given a finite Borel measure $\mu$ on $S^{n-1}$, what are the necessary and sufficient conditions on $\mu$ does there exist a convex body $K$ in $\mathbb{R}^{n}$ such that $L_{p}$-Gaussian surface area measure $S_{p,\gamma_{n},K}(\cdot\,)=\mu(\cdot\,)$? If $K$ exist, is it unique?

The first assertion involves the uniqueness result, it is known to us that the Gaussian surface area’s density $f_{r}$ of $B_{r}$ tends to $0$ as $r\rightarrow 0$ or $r\rightarrow\infty,$ this means that Gaussian surface area is may not unique. Our study in the restricted context that the Gaussian volume grater than or equal to $\frac{1}{2}$, then obtain the following uniqueness result. The same fact was firstly found in [30] in the situation of $p=1$.

Aleksandrov’s variational method implies the existence of solution with lagrange multipliers.

Due to the lack of homogeneity of measure, just like all previous Orlicz-Minkowski problems, in the solving process, the lagrange multipliers produced by Aleksandrov’s variational method can not eliminated. Recently, the work of Huang et al. in [30], the first one overcame this obstruct. Inspired their work, we obtain the existence of smooth solution.

Finally, the existence of weak solution of the $L_{p}$-Gaussian Minkowski problem follows by an approximation method with the help of a uniform estimate.

We overcome the most important obstacle of the $L_{p}$ Gaussian Minkowski problem about whether the solution degenerates or not, and obtain that it holds for asymmetric convex bodies see Theorem 3.3. But the limitation of this item is the lack of $L_{p}$ Gaussian isoperimetric type inequality for general convex body such that it can only achieve the $o$-symmetry convex body, see Theorem 4.7 and Theorem 5.5.

The paper is organized into five sections: we list some conceptual knowledge of convex bodies in Preliminaries part, and gain the key $L_{p}$-variational formula in Section 2. In Section 3, Aleksandrov’s variational method is applied to obtain the non-symmetry solution with lagrange multipliers for $p>0$. The uniqueness of solution is proved in Section 4. Section 5 provides the existence result of $o$-symmetry smooth solution for $p\geq 1$ by means of the degree theory of partial differential equation, then the $o$-symmetry weak solution is obtained with the help of approximation argument.

In this section, we make a brief review of some relevant notions about convex bodies. More detailed information of convex body theory can be found in the Schneider [44].

Let $\mathbb{R}^{n}$ represent the $n$-dimensional Euclidean space, $S^{n-1}$ be the unit sphere in $\mathbb{R}^{n},$ $C^{+}(S^{n-1})$ be positive functions defined on $S^{n-1}$ and $|\mu|$ denote total measure of the Borel measure $\mu$ on $S^{n-1}.$

Convex body in $\mathbb{R}^{n}$ is compact convex set with nonempty interior denoted by $\mathcal{K}^{n}$. Let $\partial K$ denote the boundary of $K$, $\mathcal{K}^{n}_{o}$ be convex bodies that contain the origin in interior, and $\mathcal{K}^{n}_{e}$ denote the $o$-symmetry convex bodies.

The support function $h_{K}$ of a compact convex subset $K$ in $\mathbb{R}^{n}$ is defined as

for $y\in\mathbb{R}^{n}$, where $\cdot$ represents the inner product.

For $K\in\mathcal{K}^{n}_{o}$, the radial function $\rho_{K}$ is defined by

for $x\in\mathbb{R}^{n}\setminus\{0\}$. A simple observation allows us to obtain

In fact, $\mathcal{K}^{n}$ is the space of compact convex set in $\mathbb{R}^{n}$ and it can be equipped with Hausdorff metric which associated norm is

We can easily show that $\delta(K,L)$ is the distance between $K,L\subset\mathcal{K}^{n}$. Therefore, the convergence of compact convex set sequence can be characterized by the convergence of support function. That is to say, $K_{i}\in\mathcal{K}^{n}\rightarrow K\in\mathcal{K}^{n}$ if

as $i\rightarrow\infty.$ Or if $K_{i}\in\mathcal{K}^{n}_{o}$,

as $i\rightarrow\infty,$ then $K_{i}\rightarrow K$.

Let $h\in C^{+}(S^{n-1})$, the Wulff shape $[h]$ generated by $h$ is a convex body defined by

Obviously, if $K\in\mathcal{K}^{n}_{o}$,

Let $\rho\in C^{+}(S^{n-1})$, the convex hull $\langle\rho\rangle$ generated by $\rho$ is a convex body defined by

Clearly, if $K\in\mathcal{K}^{n}_{o}$,

For $K,L\subset\mathcal{K}^{n}$ and real $a,b>0$, the Minkowski combination, $aK+bL\subset\mathcal{K}^{n}$ defined as follow

and support function is

Firey was the first to define the $L_{p}$-combination of $K,L\in\mathcal{K}^{n}_{o}$, for $a,b>0,$ $p\neq 0,$

and

Here $a\cdot K=a^{\frac{1}{p}}K$.

In case of $p\geq 1,$ $\left(a{h_{K}(v)}^{p}+b{h_{L}(v)}^{p}\right)^{\frac{1}{p}}$ is just the support function, but for $0<p<1$ is not necessarily.

For $K\in\mathcal{K}^{n}_{o}$, its polar body $K^{*}\in\mathcal{K}^{n}_{o}$ in $\mathbb{R}^{n}$ is defined by

It is easy to show that, if $K\in\mathcal{K}^{n}_{o}$,

for $x\in\mathbb{R}^{n}\setminus\{0\}$. In addition, we can get $(K^{*})^{*}=K$ if $K\in\mathcal{K}^{n}_{o}$.

For $K,K_{i}\in\mathcal{K}^{n}_{o}$, by (2.1) and the previous characterization of convex body convergence, then $K_{i}\rightarrow K$ if and only if $K_{i}^{*}\rightarrow K^{*}.$

For convex body $K\in\mathbb{R}^{n}$, then its supporting hyperplane with outward unit normal vector $v\in S^{n-1}$ represented by

The boundary point of $K$ only has one supporting hyperplane called regularity point, otherwise, it is a singular point. The set of singular points denoted as $\sigma_{K}$, it is well known that $\sigma_{K}$ has spherical Lebesgue measure $0$ (see Schneider [44], p.84).

For $x\in{\partial K\setminus\sigma_{K}}$, its Gauss map $\nu_{K}$ is represented by

Correspondingly, for Borel set $\eta\subset S^{n-1}$, its reverse Gauss map denoted by $\nu_{K}^{-1}$,

For Borel set $\eta\subset S^{n-1}$, its surface area measure is defined as

where $\mathcal{H}^{n-1}$ is $(n-1)$-dimensional Hausdorff measure.

Then $g\in C(S^{n-1}),$

For Borel set $\omega\subset S^{n-1}$, $\boldsymbol{\alpha}_{K}(\omega)$ denotes its radial Gauss image and is defined as

for $u\in\omega$. When Borel set $\omega$ have only one element $u$, we will abbreviate $\boldsymbol{\alpha}_{K}(\{u\})$ as $\boldsymbol{\alpha}_{K}(u)$. The subset of $S^{n-1}$ which make $\boldsymbol{\alpha}_{K}(u)$ contain more than one element denoted by $\omega_{K}$ for each $u\in\omega$. The set $\omega_{K}$ has spherical Lebesgue measure $0$ (see Schneider [44], Theorem 2.2.5).

The radial Guass map of $K$ is a map denoted by $\alpha_{K}(u)$, the only difference between $\alpha_{K}(u)$ and $\boldsymbol{\alpha}_{K}(u)$ is that the former is defined on $S^{n-1}\setminus\omega_{K}$ not on $S^{n-1}$ which lead to $\boldsymbol{\alpha}_{K}(u)$ may have many elements but $\alpha_{K}(u)$ has only one. In other words, if $\boldsymbol{\alpha}_{K}(u)=(\{v\})$, then $\boldsymbol{\alpha}_{K}(u)=\alpha_{K}(u)$.

For $\mathcal{H}^{n-1}$-integrable function $g:\partial K\rightarrow\mathbb{R}$,

where $F$ is defined $\mathcal{H}^{n-1}$-a.e. on $S^{n-1}$ by

For Borel set $\eta\subset S^{n-1}$, its reverse radial Gauss image $\boldsymbol{\alpha}^{*}_{K}(\eta)$ is shown as

for $v\in\eta$.

When Borel set $\eta$ have only one element $v$, we will abbreviate $\boldsymbol{\alpha}^{*}_{K}(\{v\})$ as $\boldsymbol{\alpha}^{*}_{K}(v)$. The subset of $S^{n-1}$ which make $\boldsymbol{\alpha}^{*}_{K}(v)$ contain more than one element denoted by $\eta_{K}$ for each $v\in\eta$. The spherical Lebesgue measure of set $\eta_{K}$ is $0$ (see Schneider [44], Theorem 2.2.11).

The reverse radial Gauss map of $K$ is a map denoted by $\alpha^{*}_{K}(v)$, if $\boldsymbol{\alpha}^{*}_{K}(v)=(\{u\})$, then $\boldsymbol{\alpha}^{*}_{K}(v)=\alpha^{*}_{K}(v)$.

For this part, we give the definition of the $L_{p}$-Gaussian surface area measure as follows.

The above definition is deduced from the $L_{p}$-variational formula, and the proof of variational formula requires the following lemma. Our proof mainly relies on ideas developed in [29].

The following $L_{p}$-variational formula, as we shall see in the next section, is the key to solve the $L_{p}$-Gaussian Minkowski problem.

By the definition of $S_{p,\gamma_{n},K}$, together with the conclusions that the Gauss surface area measure is weakly convergent and is absolutely continuous which were got in [30], it sufficient to reveal that $S_{p,\gamma_{n},K}$ is weakly convergent with respect Hausdorff metric, and is absolutely continuous with respect to surface area measure.

This section aims to transform the $L_{p}$-Gaussian Minkowski problem into an optimization problem by employing variational method, and prove the optimizer is just the solution to the Minkowski problem of $S_{p,\gamma_{n},K}$.