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Stability of Minkowski inequality for nearly spherical sets

Yi Wang, Shuhan Yang

TL;DR

The paper analyzes the stability of Minkowski-type inequalities for quermassintegrals on domains that are $C^1$-close to the ball, extending Glaudo’s mean-curvature stability results to fully nonlinear curvature functionals $\sigma_k(h)$. It develops a low–high frequency decomposition and leverages a suite of combinatorial identities to control curvature terms, proving stability inequalities for both general $C^1$ perturbations and axially symmetric perturbations, while also showing, via a counterexample, that removing the negative-part compensation can invalidate the inequalities. The results provide quantitative, near-sphere stability bounds for the integral of $\sigma_k(h)$ (with negative-part corrections) and demonstrate the necessity of these corrections in the nonlinear setting. Overall, the work advances the understanding of sharp Minkowski-type inequalities in nonconvex near-spherical geometries and lays groundwork for further nonlinear curvature stability results.

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 $σ_k$ 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.

Stability of Minkowski inequality for nearly spherical sets

TL;DR

The paper analyzes the stability of Minkowski-type inequalities for quermassintegrals on domains that are -close to the ball, extending Glaudo’s mean-curvature stability results to fully nonlinear curvature functionals . It develops a low–high frequency decomposition and leverages a suite of combinatorial identities to control curvature terms, proving stability inequalities for both general perturbations and axially symmetric perturbations, while also showing, via a counterexample, that removing the negative-part compensation can invalidate the inequalities. The results provide quantitative, near-sphere stability bounds for the integral of (with negative-part corrections) and demonstrate the necessity of these corrections in the nonlinear setting. Overall, the work advances the understanding of sharp Minkowski-type inequalities in nonconvex near-spherical geometries and lays groundwork for further nonlinear curvature stability results.

Abstract

In this paper, we study the stability of Minkowski inequality for nearly spherical domains that are close to the ball. We show the stability inequalities between the positive part of the curvature integrals for perturbations of a ball; we also establish the stability inequalities for axially symmetric 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.

Paper Structure

This paper contains 12 sections, 19 theorems, 196 equations.

Table of Contents

  1. Introduction
  2. Organization of the Paper.
  3. Preliminaries
  4. Important Lemmas
  5. Proofs under an additional assumption
  6. The $\sigma_2$ case
  7. The $\sigma_k$ case: Proof of Theorem \ref{['thm:4.5']}
  8. Proof of Theorem \ref{['thm:5.1']}
  9. The Proof of Theorem \ref{['thm:5.2']}
  10. The Proof of Theorem \ref{['thm:3']}
  11. The Proof of Theorem \ref{['thm:5.3']}
  12. Counterexample

Key Result

Theorem 1.2

Suppose $\Omega=\left\{\left(1+u\left(\frac{x}{|x|}\right)\right) x: x \in B\right\} \subseteq \mathbb{R}^{n+1}$ where $u \in C^{3}(\partial B)$, $\operatorname{Vol}(\Omega)=\operatorname{Vol}(B)$, and $\operatorname{bar}(\Omega)=0$. Assume $n \geq 5$, $\|u\|_{C^2}\leq M$ for some fixed $M>0$. Then, for some constant $d(k,n)>0$.

Theorems & Definitions (30)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Definition 2.1
  • ...and 20 more