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Limiting shape of the $L_p$-Minkowski problem

Shi-Zhong Du, Xu-Jia Wang, Baocheng Zhu

TL;DR

The paper investigates the asymptotic shapes of solutions to the $L_p$-Minkowski and $L_p$ dual Minkowski problems in high dimensions through a group-invariant approach based on a spanning property. It proves the existence of $\mathcal{S}$-invariant solutions for negative large $p$, and shows that the associated convex bodies converge to group-invariant polytopes tangential to $\mathbb{S}^n$, with regular polytopes arising as limiting shapes. The authors extend these results to the dual problem, establishing analogous limit behaviors: for certain parameters the limit is a polytope containing the unit ball, while in other regimes it collapses to a ball, and they prove local maximality results for polytopal limits. The findings provide a high-dimensional generalization of Andrews’ planar results, elucidate the geometric structure of limiting shapes, and enrich the theory of variational characterizations for Minkowski-type problems with symmetry constraints.

Abstract

Ben Andrews classified the limiting shape for isotropic curvature flow corresponding to the solutions of the $L_p$-Minkowski problem as $p\to-\infty$ in the planar case. In this paper, we use the group-invariant method to study the asymptotic shape of solutions to the $L_p$-Minkowski problem as $p\to-\infty$ in high dimensions. For any regular polytope $T$, we establish the existence of a solution ${Ω^{(p)}}$ to the $L_p$-Minkowski problem that converges to $T$ as $p\to-\infty$, thereby revealing the intricate geometric structure underlying this limiting behavior. We also extend the result to the dual Minkowski problem.

Limiting shape of the $L_p$-Minkowski problem

TL;DR

The paper investigates the asymptotic shapes of solutions to the -Minkowski and dual Minkowski problems in high dimensions through a group-invariant approach based on a spanning property. It proves the existence of -invariant solutions for negative large , and shows that the associated convex bodies converge to group-invariant polytopes tangential to , with regular polytopes arising as limiting shapes. The authors extend these results to the dual problem, establishing analogous limit behaviors: for certain parameters the limit is a polytope containing the unit ball, while in other regimes it collapses to a ball, and they prove local maximality results for polytopal limits. The findings provide a high-dimensional generalization of Andrews’ planar results, elucidate the geometric structure of limiting shapes, and enrich the theory of variational characterizations for Minkowski-type problems with symmetry constraints.

Abstract

Ben Andrews classified the limiting shape for isotropic curvature flow corresponding to the solutions of the -Minkowski problem as in the planar case. In this paper, we use the group-invariant method to study the asymptotic shape of solutions to the -Minkowski problem as in high dimensions. For any regular polytope , we establish the existence of a solution to the -Minkowski problem that converges to as , thereby revealing the intricate geometric structure underlying this limiting behavior. We also extend the result to the dual Minkowski problem.
Paper Structure (9 sections, 27 theorems, 119 equations)

This paper contains 9 sections, 27 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Discrete subgroup and group invariant polytope
  3. Notations
  4. Discrete subgroup and group invariant polytope
  5. Existence of group invariant solutions of \ref{['e1.2']}
  6. Limiting extremal bodies
  7. Local maximizer of the functional and proof of Theorem \ref{['t1.2']}
  8. The $L_p$ dual Minkowski problem
  9. The proof of Theorem \ref{['t1.4']}

Key Result

Theorem 1.1

Let ${\mathcal{S}}$ be a discrete subgroup of $O(n+1)$ satisfies the spanning property. Then for $p<-n-1$, there exists an ${\mathcal{S}}$-invariant solution ${h^{(p)}}$ to the $L_p$-Minkowski problem e1.2, such that the associated convex body ${\Omega^{(p)}}$ sub-converges to a group invariant poly

Theorems & Definitions (55)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.1
  • proof
  • Definition 2.1
  • Definition 2.2
  • Example 2.1
  • Theorem 3.1
  • ...and 45 more