Area, width, and logarithmic capacity of convex sets.
type: Item
instance of: scholarly article
title: Area, width, and logarithmic capacity of convex sets.
zbMATH Open document ID: 1047.30016
author: Roger W. Barnard
author: Kent Pearce
author: A. Yu. Solynin
published in: Pacific Journal of Mathematics
publication date: 2004-09-16T00:00:00Z
review text: This paper continues the study of geometric properties of the logarithmic capacity cap\(E\) of a planar compact set \(E\); (see also the paper [Ann. Acad. Sci. Fenn. Math. 27, 419--436 (2002; Zbl 1017.30033)], by the same authors). The authors prove that if cap\,\(E=1\) and the projection of \(E\) on a straight line has length \(2h\) (\(0\leq h\leq 2\)), then the area of \(E\) has an upper bound in terms of \(h\). This bound involves elliptic integrals of the second kind. The authors give a complete description of the extremal configurations in terms of certain conformal maps. They apply this result to the case when \(E\) is convex and obtain an upper bound of the area of convex sets in terms of their logarithmic capacity and their width.
reviewed by: Dimitrios Betsakos
Mathematics Subject Classification ID: 30C85
Mathematics Subject Classification ID: 31A15
Mathematics Subject Classification ID: 52A38
zbMATH DE Number: 2101119
zbMATH Keywords: logarithmic capacity
zbMATH Keywords: convex set
zbMATH Keywords: area
zbMATH Keywords: width
zbMATH Keywords: conformal map
zbMATH Keywords: symmetrization
MaRDI profile type: Publication
full work available at URL: pjm.2003.212.13
OpenAlex ID: W2060944464
DOI: 10.2140/PJM.2003.212.13
description: scientific article; zbMATH DE number 2101119