Capacity in subanalytic geometry
type: Item
instance of: scholarly article
title: Capacity in subanalytic geometry
zbMATH Open document ID: 1092.31004
author: Tobias Kaiser
published in: Illinois Journal of Mathematics
publication date: 2006-03-28T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 31B15
Mathematics Subject Classification ID: 32B20
Mathematics Subject Classification ID: 31C40
zbMATH DE Number: 5015528
zbMATH Keywords: subanalytic set
zbMATH Keywords: capacity
zbMATH Keywords: volume density
zbMATH Keywords: fine topology
zbMATH Keywords: capacitary density
reviewed by: Stephen J. Gardiner
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5015528