General extensions of an inequality due to Rogozin concerning the essential supremum of a convolution of several probability density functions on the real line are achieved. A very general theorem can be stated in the very general context of metric measure spaces, but at the cost of a more complicated formulation. By using notions of rearrangement, we prove direct generalizations of Rogozin’s original inequality to arbitrary unimodular locally compact groups for 2 summands, and to Euclidean spaces for arbitrary number of summands. By further combining with geometric inequalities related to the cube slicing results of Ball and Brzezinski, we obtain a unification and sharpening of both the infinity-Rényi 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-Vershynin (matching the sharp improvement of Livshyts-Paouris-Pivovarov).
Prof. Madiman did his B.Tech. (Elec. Engg.) from IIT Bombay and Ph.D. (Applied Math.) from Brown Uni., USA. After working for several years with Yale University, he is now with Uni. of Delaware. His research interests are information theoretic inequalities (notably the entropy power inequality) and interplay between high dimensional geometry and probability / information theory.