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The Renyi Capacity and Center

Renyi's information measures-the Renyi information, mean, capacity, radius, and center-are analyzed relying on the elementary properties of the Renyi divergence and the power means. The van Erven-Harremoes conjecture is proved for any positive order and for any set of probability measures on a given measurable space and a generalization of it is established for the constrained variant of the problem. The finiteness of the order alpha Renyi capacity is shown to imply the continuity of the Renyi capacity on (0, alpha] and the uniform equicontinuity of the Renyi information, both as a family of functions of the order indexed by the priors and as a family of functions of the prior indexed by the orders.