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
