Limiting shape of the Lₚ-Minkowski problem

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

Paper · Jan 24, 2025

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

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

math.APWider flowsarXiv:2501.14226

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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.

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Submitted: Jan 24, 2025
Subjects: math.AP, math.DG

Other curvature flows Gauss curvature flow

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

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Limiting shape of the Lp​-Minkowski problem

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