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Date
2023-07-04Type
- Working Paper
ETH Bibliography
yes
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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 $$ P(E) - P(F) \ge c\,|EΔF|, \qquad c>0. $$ Here we prove that, when $n\ge 3$, there exists a convex set $F$, with $|E|=|F|$, such that $$ P(E) - P(F) \ge c(n) \,f\big(|EΔF|\big), \qquad c(n)>0,\qquad f(t)=\frac{t}{|\log t|} \text{ for }t \ll 1. $$ 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. Show more
Permanent link
https://doi.org/10.3929/ethz-b-000643662Publication status
publishedJournal / series
arXivPages / Article No.
Publisher
Cornell UniversityEdition / version
v1Subject
Optimization and Control (math.OC); FOS: Mathematics; 49Q10, 49Q20Organisational unit
09565 - Figalli, Alessio / Figalli, Alessio
Funding
721675 - Regularity and Stability in Partial Differential Equations (EC)
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ETH Bibliography
yes
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