Rogozin's convolution inequality for locally compact groups

TL;DR

This paper extends Rogozin's convolution inequality to general Polish groups and  spaces, unifying and sharpening entropy power inequalities and bounds on measure marginals with elementary proofs.

Contribution

It generalizes Rogozin's inequality to Polish  groups and  Euclidean spaces, linking convolution bounds with entropy and measure projection inequalities.

Findings

Unified entropy power inequalities for sums of independent vectors.

Sharp bounds on marginals of product measures.

Elementary proof using extreme point characterization.

Abstract

General extensions of an inequality due to Rogozin, concerning the essential supremum of a convolution of probability density functions on the real line, are obtained. While a weak version of the inequality is proved in the very general context of Polish $\sigma$-compact groups, particular attention is paid to the group \(\mathbb{R}^d\), where the result can combined with rearrangement inequalities for certain linear images for a strong generalization. As a consequence, we obtain a unification and sharpening of both the \(\infty\)-Renyi entropy power inequality for sums of independent random vectors, due to Bobkov and Chistyakov, and the bounds on marginals of projections of product measures due to Rudelson and Vershynin (matching and extending the sharp improvement of Livshyts, Paouris and Pivovarov). The proof is elementary and relies on a characterization of extreme points of a class of probability measures in the general setting of Polish measure spaces, as well as the development of a generalization of Ball's cube slicing bounds for products of \(d\)-dimensional Euclidean balls (where the "co-dimension 1" case had been recently settled by Brzezinski).