Capacity in subanalytic geometry (Q819203)

A semianalytic set is locally given by a finite number of equalities and inequalities involving analytic functions. A subanalytic set is locally given by a projection of a relatively compact semianalytic set. This paper is concerned with the Newtonian (if \(n\geq 3\)) or logarithmic (if \(n=2\)) capacity \(c(E)\) of subanalytic sets \(E\subset \mathbb{R}^{n}\) \((n\geq 2)\). Theorem 1 says that the capacity of such a set coincides with the capacity of its closure. In the case of subanalytic sets \( E\) in the plane it is deduced that the capacitary density \( \lim_{r\rightarrow 0}c(E\cap B_{r}(x))/c(B_{r}(x))\) exists for all \(x\), where \(B_{r}(x)\) denotes the open disc of centre \(x\) and radius \(r\). For subanalytic sets in any dimension the author also establishes connections between capacitary density and certain volume densities, and between volume densities and the fine topology.