Entropic exercises around the Kneser-Poulsen conjecture

TL;DR

This paper introduces an information-theoretic framework to analyze the Kneser-Poulsen conjecture, exploring how Rényi entropies behave under contractions, and provides affirmative results in several cases.

Contribution

It presents a novel approach linking discrete geometry with information theory, specifically Rényi entropies, to address the Kneser-Poulsen conjecture.

Findings

Rényi entropies of independent sums decrease under contraction in certain cases

The approach offers new insights into geometric contraction phenomena

Connections established between entropy measures and geometric conjectures

Abstract

We develop an information-theoretic approach to study the Kneser--Poulsen conjecture in discrete geometry. This leads us to a broad question regarding whether R\'enyi entropies of independent sums decrease when one of the summands is contracted by a $1$-Lipschitz map. We answer this question affirmatively in various cases.