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An Information-Theoretic Route to Isoperimetric Inequalities via Heat Flow and Entropy Dissipation

Amandip Sangha

Abstract

We develop an information-theoretic approach to isoperimetric inequalities based on entropy dissipation under heat flow. By viewing diffusion as a noisy information channel, we measure how mutual information about set membership decays over time. This decay rate is shown to be determined by the boundary measure of the set, leading to a new proof of the Euclidean isoperimetric inequality with its sharp constant. The method extends to Riemannian manifolds satisfying curvature-dimension conditions, yielding Levy-Gromov and Gaussian isoperimetric results within a single analytic principle. Quantitative and stability bounds follow from refined entropy inequalities linking information loss to geometric rigidity. The approach connects geometric analysis and information theory, revealing how entropy dissipation encodes the geometry of diffusion and boundary.

An Information-Theoretic Route to Isoperimetric Inequalities via Heat Flow and Entropy Dissipation

Abstract

We develop an information-theoretic approach to isoperimetric inequalities based on entropy dissipation under heat flow. By viewing diffusion as a noisy information channel, we measure how mutual information about set membership decays over time. This decay rate is shown to be determined by the boundary measure of the set, leading to a new proof of the Euclidean isoperimetric inequality with its sharp constant. The method extends to Riemannian manifolds satisfying curvature-dimension conditions, yielding Levy-Gromov and Gaussian isoperimetric results within a single analytic principle. Quantitative and stability bounds follow from refined entropy inequalities linking information loss to geometric rigidity. The approach connects geometric analysis and information theory, revealing how entropy dissipation encodes the geometry of diffusion and boundary.

Paper Structure

This paper contains 32 sections, 12 theorems, 195 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Geometric measure and perimeter
  4. Heat semigroup and diffusion
  5. Entropy and mutual information
  6. Communication, entropy, and thermodynamics.
  7. The Heat Flow as an Information Channel
  8. The noisy--label model
  9. Mutual information under diffusion
  10. Information dissipation identity
  11. Entropy Dissipation and Boundary Geometry
  12. Small--time expansion of conditional entropy
  13. Entropy dissipation law
  14. An information–theoretic reformulation of the isoperimetric problem
  15. Explicit derivation in Euclidean space
  16. ...and 17 more sections

Key Result

Proposition 3.1

We have the information dissipation identity

Theorems & Definitions (24)

  • Proposition 3.1
  • proof
  • Theorem 4.1: Heat-content and $L^1$--jump asymptotics
  • proof
  • Corollary 4.2: Entropy boundary-layer asymptotics
  • proof
  • Corollary 4.3: Information–theoretic surface area
  • proof
  • Theorem 4.4: Entropy dissipation and perimeter
  • proof
  • ...and 14 more