Stability of Minkowski inequality for nearly spherical sets

TL;DR

This paper investigates the stability of Minkowski inequalities for nearly spherical domains, establishing new inequalities for curvature integrals under perturbations and highlighting limitations with counterexamples.

Contribution

It extends stability results of Minkowski inequalities to fully nonlinear curvature cases for nearly spherical sets, including symmetric perturbations.

Findings

Stability inequalities hold for $C^1$ close to spherical domains.

Inequalities are valid for axially symmetric $C^1$ perturbations.

Counterexamples show inequalities fail without compensating for negative curvature parts.

Abstract

In this paper, we study the stability of Minkowski inequality for nearly spherical domains that are $C^1$ close to the ball. We show the stability inequalities between the positive part of the $\sigma_k$ curvature integrals for $C^1$ perturbations of a ball; we also establish the stability inequalities for axially symmetric $C^1$ perturbations of a ball. Finally, we construct a counterexample, illustrating that the inequalities become invalid if we do not compensate the integral with the negative part of the curvature. Our work generalizes Glaudo's results on the mean curvature integral to the fully nonlinear cases.