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

TL;DR

This paper introduces an information-theoretic method to prove isoperimetric inequalities by analyzing entropy dissipation during heat flow, unifying geometric and probabilistic perspectives across Euclidean and Riemannian settings.

Contribution

It presents a novel entropy-based approach to derive sharp isoperimetric inequalities and extends these results to curved spaces under curvature-dimension conditions.

Findings

Provides a new proof of the Euclidean isoperimetric inequality with sharp constant

Extends the approach to Riemannian manifolds satisfying curvature-dimension conditions

Establishes quantitative and stability bounds through refined entropy inequalities

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.