Limiting shape of the $L_p$-Minkowski problem

TL;DR

This paper investigates the asymptotic geometric behavior of solutions to the $L_p$-Minkowski problem as p approaches negative infinity, revealing convergence to regular polytopes in high dimensions.

Contribution

It extends Andrews' planar results to high dimensions using group-invariant methods, showing solutions converge to regular polytopes as p approaches -infinity.

Findings

Solutions to the $L_p$-Minkowski problem converge to regular polytopes as p→-∞

Existence of solutions converging to any regular polytope in high dimensions

Results extend to the dual Minkowski problem

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 ${\Omega^{(p)}}$ 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.