Strong stability of convexity with respect to the perimeter

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.