Neighbourhood capacities

Abstract

We study the behaviour of the p-capacity of a compact set E with respect to the t-neighbourhoods of E as t varies. We establish sharp upper and lower bounds for these capacities in terms of Minkowski and Hausdorff type contents of E, respectively, and our results hold in both Euclidean and more general metric spaces. In our lower bounds the porosity of the set E plays an important role, and it is shown by examples that an assumption like this is in general necessary. In addition, we present a self-contained approach to the theory of sets of zero capacity in metric spaces.