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 $α$ Renyi capacity is shown to imply the continuity of the Renyi capacity on $(0,α]$ 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. The Renyi capacities and centers of various families of Poisson processes are derived as examples.