The Kneser--Poulsen phenomena for entropy

TL;DR

This paper proves an information-theoretic version of the Kneser--Poulsen conjecture, showing that entropy does not increase when probability measures are contracted, and connects this to Gaussian convolutions and heat flow.

Contribution

It establishes the validity of the entropy-based Kneser--Poulsen phenomenon and provides a unified inequality linking entropy power and Gaussian convolutions.

Findings

Entropy comparisons are preserved under heat flow for contracted measures.

The entropy power concavity result is unified with the entropic Kneser--Poulsen theorem.

The paper confirms the conjecture's analogue in an information-theoretic setting.

Abstract

The Kneser--Poulsen conjecture asserts that the volume of a union of balls in Euclidean space cannot be increased by bringing their centres pairwise closer. We prove that its natural information-theoretic counterpart is true. This follows from a complete answer to a question asked in arXiv:2210.12842 about Gaussian convolutions, namely that the R\'enyi entropy comparisons between a probability measure and its contractive image are preserved when both undergo simultaneous heat flow. An inequality that unifies Costa's result on the concavity of entropy power with the entropic Kneser--Poulsen theorem is also presented.