Capacity and Covering Numbers

We establish the inequality 1/C(K)(E) >= integral(infinity)(0) |d(K(t))|/N(E)(t), where E is a compact metric space, K is a kernel function, C(K) is the associated capacity, and NE( 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 delta-neighborhoods of a set. We also investigate possible converses to the inequality.