Isoperimetric inequalities in convex cylinders and cylindrically bounded convex bodies

Ritoré, Manuel; Vernadakis, Efstratios

doi:10.1007/s00526-014-0800-3

Cited by 11 publications

(8 citation statements)

References 25 publications

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“…In part of this work, we focus on convex cones of R n and our motivation is due to the existence of a lot of interest in quantitative estimates for isoperimetric inequalities and existence of isoperimetric regions in such setting. Relevant contributions in this direction were obtained by several mathematicians including P. L. Lions, F. Pacella, J. Choe, F. Morgan, M. Ritoré, C. Rosales, A. Figali and E. Indrei (see, for instance, [19,4,24,27,28,6] and references therein, as well as relevant discussion in [26,17]).…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

Capacity inequalities and rigidity of cornered/conical manifolds

Tiarlos

2018

Ann Glob Anal Geom

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We prove capacity inequalities involving the total mean curvature of hypersurfaces with boundary in convex cones and the mass of asymptotically flat manifolds with non-compact boundary. We then give the analogous of Pölia-Szegö, Alexandrov-Fenchel and Penrose type inequalities in this setting. Among the techniques used in this paper are the inverse mean curvature flow for hypersurfaces with boundary.Key words and phrases. Capacity, inverse mean curvature flow, rigidity, Riemannian penrose inequality, convex cone.This work was completed with the support of Cnpq/Brazil.

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Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

Capacity inequalities and rigidity of cornered/conical manifolds

Tiarlos

2018

Ann Glob Anal Geom

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“…In the case of unbounded convex bodies, several results on the isoperimetric profile of cylindrically bounded convex bodies have been obtained in [69] and for conically bounded ones in [72]. In convex cones, the results by Lions and Pacella [43] were recovered by Ritoré and Rosales [68] using stability techniques.…”

Section: Historical Backgroundmentioning

confidence: 99%

Isoperimetric inequalities in unbounded convex bodies

Leonardi¹,

Ritoré²,

Vernadakis³

2016

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Contents Chapter 1. Introduction 1.1. Historical background 1.2. Outline of contents Chapter 2. Convex bodies and finite perimeter sets 2.1. Convex bodies and local convergence in Hausdorff distance 2.2. Finite perimeter sets and isoperimetric profile Chapter 3. Unbounded convex bodies of uniform geometry 3.1. Asymptotic cylinders 3.2. Convex bodies of uniform geometry 3.3. Density estimates and a concentration lemma 3.4. Examples Chapter 4. A generalized existence result 4.1. Preliminary results 4.2. Proof of the main result Chapter 5. Concavity of the isoperimetric profile 5.1. Continuity of the isoperimetric profile 5.2. Approximation by smooth sets 5.3. Concavity of the isoperimetric profile Chapter 6. Sharp isoperimetric inequalities and isoperimetric rigidity 6.1. Convex bodies with non-degenerate asymptotic cone 6.2. The isoperimetric profile for small volumes 6.3. Isoperimetric rigidity Chapter 7. The isoperimetric dimension of an unbounded convex body 7.1. An asymptotic isoperimetric inequality 7.2. Estimates on the volume growth of balls 7.3. Examples Bibliography iii iv CONTENTS

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“…2.11]. As observed by Morgan, an alternative proof for k = 1 can be given using the monotonicity formula and properties of the isoperimetric profile of M × (see [20,Cor. 4.12] for a proof when M is a convex body).…”

Section: Introductionmentioning

confidence: 96%