Strong stability of convexity with respect to the perimeter

TL;DR

This paper establishes a quantitative stability estimate for convexity with respect to perimeter in higher dimensions, showing that deviations from convexity are controlled by a specific logarithmic function of the symmetric difference.

Contribution

It proves a new stability inequality for convex sets in dimensions three and higher, involving a logarithmic correction, and relates it to Alexandrov's Theorem on constant mean curvature sets.

Findings

The stability estimate holds with a logarithmic correction term.

The inequality fails for a linear function of the symmetric difference.

The result applies to small $C^2$-deformations of the unit ball.

Abstract

Let $E\subset \mathbb R^n$, $n\ge 2$, be a set of finite perimeter with $|E|=|B|$, where $B$ denotes the unit ball. When $n=2$, since convexification decreases perimeter (in the class of open connected sets), it is easy to prove the existence of a convex set $F$, with $|E|=|F|$, such that $$ P(E) - P(F) \ge c\,|E\Delta F|, \qquad c>0. $$ Here we prove that, when $n\ge 3$, there exists a convex set $F$, with $|E|=|F|$, such that $$ P(E) - P(F) \ge c(n) \,f\big(|E\Delta F|\big), \qquad c(n)>0,\qquad f(t)=\frac{t}{|\log t|} \text{ for }t \ll 1. $$ Moreover, one can choose $F$ to be a small $C^2$-deformation of the unit ball. Furthermore, this estimate is essentially sharp as we can show that the inequality above fails for $f(t)=t.$ Interestingly, the proof of our result relies on a new stability estimate for Alexandrov's Theorem on constant mean curvature sets.