This book discusses two venues of the isoperimetric inequality: (i) the sharp inequality in Euclidean space, with characterization of equality, and (ii) isoperimetric inequalities in Riemannian manifolds, where precise inequalities are unavailable but rough inequalities nevertheless yield qualitative global geometric information about the manifolds. In Euclidean space, a variety of proofs are presented, each slightly more ambitious in its application to domains with irregular boundaries.

Preface

Ⅰ Introduction

Ⅰ.1 The Isoperimetric Problem

Ⅰ.2 The Isoperimetric Inequality in the Plane

Ⅰ.3 Preliminaries

Ⅰ.4 Bibliographic Notes

Ⅱ Differential Geometric Methods

Ⅱ.1 The C2 Uniqueness Theory

Ⅱ.2 The Cl Isoperimetric Inequality

Ⅱ.3 Bibliographic Notes

Ⅲ Minkowski Area and Perimeter

Ⅲ.1 The Hausdorff Metric on Compacta

Ⅲ.2 Minkowski Area and Steiner Symmetrization

Ⅲ.3 Application: The Faber-Krahn Inequality

Ⅲ.4 Perimeter

Ⅲ.5 Bibliographic Notes

Ⅳ Hansdorff Measure and Perimeter

Ⅳ.1 Hausdorff Measure

Ⅳ.2 The Area Formula for Lipschitz Maps

Ⅳ.3 Bibliographic Notes

Ⅴ Isoperimetric Constants

Ⅴ.1 Riemannian Geometric Preliminaries

Ⅴ.2 Isoperimetric Constants

Ⅴ.3 Discretizations and Isoperimetric Inequalities

Ⅴ.4 Bibliographic Notes

Ⅵ Analytic Isoperimetric Inequalities

Ⅵ.1 L2 Sobolev Inequalities

Ⅵ.2 The Compact Case

Ⅵ.3 Faber-Krahn Inequalities

Ⅵ.4 The Federer-Fleming Theorem:The Discrete Case

Ⅵ.5 Sobolev Inequalities and Discretizations

Ⅵ.6 Bibliographic Notes

Ⅶ Laplace and Heat Operators

Ⅷ Large Time Heat Diffusion

Bibliography

Author Index

Subject Index