Minkowski inequality for nearly spherical domains

TL;DR

This paper studies Minkowski-like inequalities for domains close to a sphere, proving sharp and near-sharp inequalities, establishing stability results, and providing counterexamples for certain variants.

Contribution

It introduces new sharp and stability Minkowski inequalities for nearly spherical domains, including the case of axial symmetry, and demonstrates limitations through counterexamples.

Findings

Proved a sharp geometric inequality involving second fundamental form for nearly spherical domains.

Established the stability of the volumetric Minkowski inequality under $C^1$ perturbations.

Provided counterexamples showing certain inequalities fail when replacing $H^+$ with $H$.

Abstract

We investigate the validity and the stability of various Minkowski-like inequalities for $C^1$-perturbations of the ball. Let $K\subseteq\mathbb R^n$ be a domain (possibly not convex and not mean-convex) which is $C^1$-close to a ball. We prove the sharp geometric inequality $$ \left(\int_{\partial K} \lVert II\rVert_1 d\mathscr H^{n-1}\right)^{\frac1{n-2}} \ge C_1(n)Per(K)^{\frac1{n-1}} , $$ where $C_1(n)$ is the constant that yields the equality when $K=B_1$ (and $\lVert II\rVert_1$ is the sum of the absolute values of the eigenvalues of the second fundamental form $II$ of $\partial K$). Moreover, for any $\delta>0$, if $K$ is sufficiently $C^1$-close to a ball, we show the almost sharp Minkowski inequality $$ \left(\int_{\partial K} H^+ d\mathscr H^{n-1}\right)^{\frac1{n-2}} \ge (C_1(n)-\delta)Per(K)^{\frac1{n-1}} . $$ If $K$ is axially symmetric, we prove the Minkowski inequality with the sharp constant (i.e., $\delta=0$). We establish also the sharp quantitative stability (in the family of $C^1$-perturbations of the ball) of the volumetric Minkowski inequality $$ \left(\int_{\partial K} H^+ d\mathscr H^{n-1}\right)^{\frac1{n-2}} \ge C_2(n)|K|^{\frac1n} , $$ where $C_2(n)$ is the constant that yields the equality when $K=B_1$. Finally, we show, by constructing a counterexample, that the mentioned inequalities are false (even for domains $C^1$-close to the ball) if one replaces $H^+$ with $H$.