Infinity-Rényi entropy power inequalities

An optimal ∞-Rényi entropy power inequality is derived for d-dimensional random vectors. In fact, the authors establish a matrix ∞-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 ∞-Rényi entropy inequalities, and then putting forth a new rearrangement inequality for the ∞-Rényi entropy. Quantitative results are then derived as consequences of a new geometric inequality for uniform distributions on Euclidean balls.