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- W4383469106 abstract "Let $Esubset mathbb R^n$, $nge 2$, 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 $$ P(E) - P(F) ge c,|EDelta F|, qquad c>0. $$ Here we prove that, when $nge 3$, there exists a convex set $F$, with $|E|=|F|$, such that $$ P(E) - P(F) ge c(n) ,fbig(|EDelta F|big), qquad c(n)>0,qquad f(t)=frac{t}{|log t|} text{ for }t ll 1. $$ 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." @default.
- W4383469106 created "2023-07-07" @default.
- W4383469106 creator A5021329629 @default.
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- W4383469106 date "2023-07-04" @default.
- W4383469106 modified "2023-10-18" @default.
- W4383469106 title "Strong stability of convexity with respect to the perimeter" @default.
- W4383469106 doi "https://doi.org/10.48550/arxiv.2307.01633" @default.
- W4383469106 hasPublicationYear "2023" @default.
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