Capacity and Covering Numbers

We establish the inequality $1/C_K(E)\ge \int_0^\infty |dK(t)|/N_E(t)$ , where E is a compact metric space, K is a kernel function, C _K is the associated capacity, and N _E (t) denotes the minimal number of sets of diameter t needed to cover E. We give applications to the capacity of generalized Cantor sets, and to the capacity of δ-neighborhoods of a set. We also investigate possible converses to the inequality.