Strong stability of convexity with respect to the perimeter

Let $E\subset {\mathbb{R}}^n$, $n\ge 2$, be a set of finite perimeter with $\mathrm{\mid }E\mathrm{\mid }=\mathrm{\mid }B\mathrm{\mid }$, 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 $\mathrm{\mid }E\mathrm{\mid }=\mathrm{\mid }F\mathrm{\mid }$, such that $$P(E)-P(F)\ge c\mathrm{\mid }E\mathrm{\Delta }F\mathrm{\mid },c>0.$$ Here we prove that, when $n\ge 3$, there exists a convex set $F$, with $\mathrm{\mid }E\mathrm{\mid }=\mathrm{\mid }F\mathrm{\mid }$, such that $$P(E)-P(F)\ge c(n)f\left(\mathrm{\mid }E\mathrm{\Delta }F\mathrm{\mid }\right),c(n)>0,f(t)=\frac{t}{\mathrm{\mid }\log t\mathrm{\mid }}\text{ for }t\ll 1.$$ 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 $f(t)=t\mathrm{.}$ Interestingly, the proof of our result relies on a new stability estimate for Alexandrov’s Theorem on constant mean curvature sets.