Infinity-Renyi Entropy Power Inequalities

An optimal infinity-Renyi entropy power inequality is derived for d-dimensional random vectors. In fact, the authors establish a matrix infinity-EPI analogous to the generalization of the classical EPI established by Zamir and Feder. The result is achieved by demonstrating uniform distributions as extremizers of a certain class of infinity-Renyi entropy inequalities, and then putting forth a new rearrangement inequality for the infinity-Renyi entropy. Quantitative results are then derived as consequences of a new geometric inequality for uniform distributions on Euclidean balls.