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

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.