Submitted Paper

Inserted: 12 jun 2024
Last Updated: 16 sep 2024

Year: 2023

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.