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Isoperimetric inequalities in unbounded convex bodies
About this Title
Gian Paolo Leonardi, Manuel Ritoré and Efstratios Vernadakis
Publication: Memoirs of the American Mathematical Society
Publication Year:
2022; Volume 276, Number 1354
ISBNs: 978-1-4704-5118-9 (print); 978-1-4704-7020-3 (online)
DOI: https://doi.org/10.1090/memo/1354
Published electronically: March 1, 2022
Keywords: Isoperimetric inequalities,
isoperimetric profile,
isoperimetric regions,
convex bodies,
asymptotic cylinders,
rigidity,
isoperimetric dimension
Table of Contents
Chapters
- List of symbols
- 1. Introduction
- 2. Convex bodies and finite perimeter sets
- 3. Unbounded convex bodies of uniform geometry
- 4. A generalized existence result
- 5. Concavity of the isoperimetric profile
- 6. Sharp isoperimetric inequalities and isoperimetric rigidity
- 7. Asymptotic behavior of the isoperimetric profile of an unbounded convex body
Abstract
We consider the problem of minimizing the relative perimeter under a volume constraint in an unbounded convex body $C\subset \mathbb {R}^{n}$, without assuming any further regularity on the boundary of $C$. Motivated by an example of an unbounded convex body with null isoperimetric profile, we introduce the concept of unbounded convex body with uniform geometry. We then provide a handy characterization of the uniform geometry property and, by exploiting the notion of asymptotic cylinder of $C$, we prove existence of isoperimetric regions in a generalized sense. By an approximation argument we show the strict concavity of the isoperimetric profile and, consequently, the connectedness of generalized isoperimetric regions. We also focus on the cases of small as well as of large volumes; in particular we show existence of isoperimetric regions with sufficiently large volumes, for special classes of unbounded convex bodies. We finally address some questions about isoperimetric rigidity and analyze the asymptotic behavior of the isoperimetric profile in connection with the notion of isoperimetric dimension.- Gabriel Acosta and Ricardo G. Durán, An optimal Poincaré inequality in $L^1$ for convex domains, Proc. Amer. Math. Soc. 132 (2004), no. 1, 195–202. MR 2021262, DOI 10.1090/S0002-9939-03-07004-7
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