Isoperimetric Inequalities in Unbounded Convex Bodies.

Format:

Book

Author/Creator:

Leonardi, Gian Paolo.

Contributor:

Ritoré, Manuel.

Vernadakis, Efstratios.

Series:

Memoirs of the American Mathematical Society

Memoirs of the American Mathematical Society ; v.276

Language:

English

Subjects (All):

Convex bodies.

Boundary value problems.

Isoperimetric inequalities.

Physical Description:

1 online resource (100 pages)

Edition:

1st ed.

Place of Publication:

Providence : American Mathematical Society, 2022.

Summary:

"We consider the problem of minimizing the relative perimeter under a volume constraint in an unbounded convex body C Rn, 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"-- Provided by publisher.

Contents:

Cover

Title page

List of symbols

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. 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. Asymptotic behavior of the isoperimetric profile of an unbounded convex body

7.1. An asymptotic isoperimetric inequality

7.2. Estimates on the volume growth of balls

7.3. Examples

Bibliography

Back Cover.

Notes:

Description based on publisher supplied metadata and other sources.

Other Format:

Print version: Leonardi, Gian Paolo Isoperimetric Inequalities in Unbounded Convex Bodies

ISBN:

9781470470203

1470470209

OCLC:

1321796883