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\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,\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.
