Limiting shape of the $L_{p}$-Minkowski problem

The classical Minkowski problem looks for a convex body such that its surface measure matches a given Radon measure $\mu$ on the unit sphere ${\mathbb{S}}^{n}$ in $\mathbb{R}^{n+1}$. This problem dates back to the early works of Minkowski, Aleksandrov and Fenchel-Jessen (see [33]). For a convex body $\Omega\in{\mathcal{K}}_{0}$ (the set of convex bodies containing the origin) and two parameters $p,q\in{\mathbb{R}}$, Lutwak,Yang and Zhang [30] introduced the measure

for any measurable set $\omega\subset{\mathbb{S}}^{n}$, and proposed the following dual Minkowski problem. Given a finite Radon measure $\mu$ on ${\mathbb{S}}^{n}$, find a convex body $\Omega\in{\mathcal{K}}_{0}$ such that $d\mathcal{C}_{p,q}(\Omega,\cdot)=d\mu$. Here, $G:\partial\Omega\rightarrow{\mathbb{S}}^{n}$ is the Gauss map, $h:{\mathbb{S}}^{n}\to{\mathbb{R}}$ is the support function of $\Omega\in\mathcal{K}_{0}$, and $\mathcal{H}^{n}$ is the $n$-dimensional Hausdorff measure. In the smooth category, the measure $\mu$ is given by $d\mu=fd\sigma$ for a positive function $f$, where $d\sigma$ is the spherical Lebesgue measure, the $L_{p}$ dual Minkowski problem can then be formulated by the equation

This is a fully nonlinear equation of Monge-Ampère type, which has received increasing attention in recent years (see e.g. [6, 11, 21, 22, 30, 35, 36]). When $q=n+1$, equation (1.1) reduces to the $L_{p}$-Minkowski problem, introduced in [28], given by

Here, $\nabla$ denotes the standard Levi-Civita connection on the sphere ${\mathbb{S}}^{n}$, and $I$ is the identity matrix.

The $L_{p}$-Minkowski problem has been extensively studied in recent years (see e.g. [1, 2, 4, 7, 8, 16, 19, 20, 23, 24, 27, 29, 31, 32, 37, 38]). According to the Blaschke-Santaló inequality, the problem is divided into three cases, the sub-critical case $p>-n-1$, the critical case $p=-n-1$, and the super-critical case $p<-n-1$. There are rich phenomena on the existence and multiplicity of solutions. In the sub-critical case $p\in(-n-1,0]$, the solution is usually not unique [25]. When $p<-n$, there may be infinitely many solutions [26]. In the critical case $p=-n-1$, due to a Kazdan-Warner type obstruction [13], there may be no solutions to the problem (1.2). In the super-critical case $p<-n-1$, surprisingly, Guang, Li and the second author [18] proved that for any $p<-n-1$ and any positive, smooth function $f$, there is a solution to (1.2). An amazing result was obtained by Andrews [3]. He proved that in the planar case $n=1$, for any regular polygon $T_{k}$ (with $k$-sides), there is a solution to (1.2) which converges to $T_{k}$, as $p$ tends to negative infinity. This raises a natural question of whether such limiting behavior holds in high dimensions:

In fact, Andrews classified the limiting shape of the solutions to (1.2) in the planar case by solving an isotropic curvature flow. In this paper, we explore a new method, called the group-invariant method, to study the limiting shape of the solutions to (1.2) in high dimensions. To do so, we assume that, throughout the paper, $f$ satisfies the following conditions:

• (i)

  $f$ is a measurable function satisfying

  (1.3)

  $$c_{1}\leq f\leq c_{2}$$

  for positive constants $c_{1},c_{2}$.

• (ii)

  $f$ is invariant with respect to a group ${\mathcal{S}}$, which is a discrete subgroup of $O(n+1)$ satisfying the spanning property (introduced in Section 2).

A particular case is when $\mathcal{S}$ is the discrete subgroup of $O(n+1)$ generated by a regular polytope.

First, we obtain the asymptotic behavior of solutions to the $L_{p}$-Minkowski problem (1.2) for negative large $p$ in high dimensions, which solves part (i) of Problem 1.1.

We say that a polytope $T$ is tangential to the unit sphere ${\mathbb{S}}^{n}$ if every face of $T$ is tangential to ${\mathbb{S}}^{n}$. It is easy to see that if $T$ is tangential to ${\mathbb{S}}^{n}$, then $T$ is convex and $T$ contains the unit ball ${B_{1}}$.

By the regularity theory of Caffarelli [9, 10], the solution $h^{(p)}$ is $C^{1,\alpha}$ if $f$ is positive and bounded, and is $C^{2,\alpha}$ if $f$ is also Hölder continuous. By the maximum principle, there is a unique solution $h^{(p)}$ to (1.2) when $p>n+1$ [13]. It is easy to see that the solution $h^{(p)}$ tends to constant one when $p$ tends to infinity. In fact, at the maximum point of $h^{(p)}$, we have

by equation (1.2). Hence $\lim_{p\to+\infty}\sup h^{(p)}=1$. Similarly, one can show that $\lim_{p\to+\infty}\inf h^{(p)}=1$ by evaluating (1.2) at the minimum point of $h^{(p)}$. We conclude that the limiting shape of the solutions $\Omega^{(p)}$ is the unit ball ${B_{1}}$ as $p\to+\infty$.

Theorem 1.1 deals with the case $p<-n-1$. When $f\equiv 1$, there is a trivial solution $h\equiv 1$ to (1.2). Theorem 1.1 tells that there is a non-trivial solution $h^{(p)}$ when $p$ is negative large. The solution $h^{(p)}$ is a maximizer of the functional

among all $\mathcal{S}$-invariant convex bodies $\Omega$ containing the origin, where $h=h_{\Omega}$ is the support function of $\Omega$,

Here the value $V(\Omega)$ is equal to $(n+1)$ times the volume $|\Omega|$. The spanning property of the group $\mathcal{S}$ guarantees the uniform estimate of group invariant solutions, and one can obtain the existence of solutions for all $p\neq n+1$. By property of the functional $\mathcal{F}_{p}$, we show that the maximizer $h^{(p)}$ is not a constant when $p$ is negative large, and converges to a limit as $p$ tends to negative infinity. Moreover, the limit is a polytope.

The next theorem shows that every regular polytope tangent to the unit sphere is a limit polytope, thereby resolving part (ii) of Problem 1.1.

A regular polytope has strong symmetry. We prove Theorem 1.2 by considering maximizers of the functional $\mathcal{F}_{p}$ among convex bodies with the strong symmetry. Theorems 1.1 and 1.2 can be regarded as a higher dimensional extension of Andrews’s result in ([3], Theorem 1.5) in the planar case. Here we allow $f$ to be a measurable function.

Next we consider the dual Minkowski problem (1.1). Under the group invariant assumptions in Theorem 1.1, we have uniform estimate for the group invariant solution $h^{(p,q)}$ to (1.1), which is a maximizer of the functional

among all $\mathcal{S}$-invariant convex bodies $\Omega\subset\mathcal{K}_{0}(\mathcal{S})$, where

and $V_{n+1}(\Omega)=V(\Omega)$.

Here we are interested in the asymptotic behaviour of the solution $h^{(p,q)}$ as $p$ tends to negative infinity for fixed $q$, and the asymptotic behaviour as $q$ tends to positive infinity for fixed $p$.

The next theorem shows that every regular polytope tangential to the unit sphere is a limit polytope.

This paper is organized as follows. In Section 2, we introduce some notations and properties concerning subgroups $\mathcal{S}\subset O(n+1)$ and the corresponding group invariant polytopes. In Section 3, we prove the existence of non-trivial group invariant solutions to (1.2) by a variational method for negative large $p$. By passing to the limit $p\to-\infty$, we characterize the limiting convex body $\Omega^{(-\infty)}$ in Section 4. In Section 5, we show that a regular polytope $T$ is a strictly local maximizer of the functional $\mathcal{F}_{p}$ and thus give a proof of Theorem 1.2. Finally, we consider the $L_{p}$ dual Minkowski problem (1.1) and prove Theorems 1.3-1.4 in Sections 6-7.

Group invariant solutions have been studied in various geometric problems. Here we consider the discrete subgroup ${\mathcal{S}}\subset O(n+1)$ that satisfies the spanning property. By the bounded ratio condition in Section 2, we can prove the following existence result.

To prove Theorem 3.1, it suffices to prove

To prove Lemma 3.1, we first prove the following lemmas.

As a corollary of Lemma 3.3, we have the following result.

When a-priori lower-upper bounds of functions $h\in{\mathcal{K}}_{p}({\mathcal{S}})$ have been derived in Lemmas 3.2 and 3.3, by Blaschke’s selection theorem, a maximizing sequence $\Omega_{j}\in{\mathcal{K}}_{p}({\mathcal{S}})$ converges to a limiting convex body ${{\Omega^{(p)}}}\in{\mathcal{K}}_{p}({\mathcal{S}})$. Due to the uniform convergence of the support functions $h_{j}$ to ${{h^{(p)}}}$, we see that ${{\Omega^{(p)}}}$ is actually a maximizer of the variational problem (3.1). Consequently, we can prove Lemma 3.1.

By Theorem 3.1 and its proof, we have

It is worth noting that when the Santaló center of $\Omega$ is the origin and when $f=1$, our functional $\mathcal{F}_{-n-1}(\cdot)$ is exactly Mahler’s volume [33]. By Lemmas 3.2 and 3.3, the sequence of maximizers ${{h^{(p)}}}$ of the functional is a-priori bounded in $Lip({\mathbb{S}}^{n})$ by convexity. Hence, along a subsequence, the convex body ${{\Omega^{(p)}}}$ converges to a limiting convex body $\Omega^{(-\infty)}\in{\mathcal{K}}_{0}({\mathcal{S}})$. We have the following characterization of $\Omega^{(-\infty)}$.

The next lemma rules out the trivial possibility of the limiting sphere ${\mathbb{S}}^{n}$.

The next lemma implies that the maximizer of the limiting functional must be a polytope.

Combining with Theorems 3.1 and 4.1, along with Lemma 4.3, we obtain the following theorem, which characterizes the limiting extremal body as a maximal group-invariant polytope.

In this section, we use $T\subset\mathbb{R}^{n+1}$ to denote a regular polytope that is tangential to the unit sphere ${\mathbb{S}}^{n}$, and $\mathcal{S}_{T}\subset O(n+1)$ to represent the symmetry group of $T$. Denote

the $\delta$-neighborhood of $T$ for a positive constant $\delta$ and the Hausdorff distance $dist(\cdot,\cdot)$, and denote $\partial{\mathcal{N}}_{\delta}(T)$ the boundary of ${\mathcal{N}}_{\delta}(T)$. Denote also the normalized group invariant family

and the normalized $\delta$-neighborhood

of $T$. Then, we can prove that $T$ has strictly maximal volume in the normalized family ${\mathcal{K}}_{min}(\delta,{\mathcal{S}}_{T})$ for some $\delta=\delta_{T}>0$ as follows.