Uniformly fat sets

In this paper we study closed sets E E which are "locally uniformly fat" with respect to a certain nonlinear Riesz capacity. We show that E E is actually "locally uniformly fat" with respect to a weaker Riesz capacity. Two applications of this result are given. The first application is concerned with proving Sobolev-type inequalities in domains whose complements are uniformly fat. The second application is concerned with the Fekete points of E E .