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The isoperimetric inequality

Simon Brendle, Michael Eichmair

Abstract

We discuss several classical and recent proofs of the isoperimetric inequality and the Sobolev inequality.

The isoperimetric inequality

Abstract

We discuss several classical and recent proofs of the isoperimetric inequality and the Sobolev inequality.
Paper Structure (5 sections, 8 theorems, 40 equations)

This paper contains 5 sections, 8 theorems, 40 equations.

Table of Contents

  1. The isoperimetric inequality and the Sobolev inequality
  2. Proof of Theorem \ref{['sobolev.inequality.on.a.ball']} using measure transportation
  3. Proof of Theorem \ref{['sobolev.inequality.on.a.ball']} using the Alexandrov-Bakelman-Pucci method
  4. The Sobolev inequality and the isoperimetric inequality on a hypersurface in $\mathbb{R}^{n+1}$
  5. Outlook: Isoperimetric problems in Riemannian geometry and mathematical relativity

Key Result

Theorem 1

Let $E$ be a compact domain in $\mathbb{R}^2$ with smooth boundary. Then

Theorems & Definitions (10)

  • Theorem 1: Isoperimetric inequality in the plane
  • Theorem 2: Isoperimetric inequality in $\mathbb{R}^n$
  • Theorem 3: Sobolev inequality on $\mathbb{R}^n$
  • Theorem 4: Sobolev inequality on a ball
  • Remark 5
  • Definition 6
  • Theorem 7: Isoperimetric inequality for minimal disks
  • Theorem 8: Sobolev inequality on a hypersurface
  • Corollary 9: Sobolev inequality on a minimal hypersurface
  • Corollary 10: Isoperimetric inequality on a minimal hypersurface