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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 convexificationdecreases perimeter (in the class of open connected sets), it is easy to provethe 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 $n\ge3$, there exists a convex set $F$, with $|E|=|F|$, such that $$  P(E) - P(F) \ge c(n) \,f\big(|E\Delta F|\big), \qquad c(n)>0,\qquadf(t)=\frac{t}{|\log t|} \text{ for }t \ll 1. $$ Moreover, one can choose $F$ tobe a small $C^2$-deformation of the unit ball. Furthermore, this estimate isessentially 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 forAlexandrov's Theorem on constant mean curvature sets.

Strong stability of convexity with respect to the perimeter