Abstract
An optimal ∞-Renyi 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 ∞-Renyi entropy inequalities, and then putting forth a new rearrangement inequality for the ∞-Renyi entropy. Quantitative results are then derived as consequences of a new geometric inequality for uniform distributions on Euclidean balls.
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