Strong stability of convexity with respect to the perimeter

Alessio Figalli, Yi Ru-Ya Zhang

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 cl…