Rogozin's convolution inequality for locally compact groups

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).