A reverse isoperimetric inequality for convex plane curves
type: Item
instance of: scholarly article
title: A reverse isoperimetric inequality for convex plane curves
zbMATH Open document ID: 1121.52023
author: Shengliang Pan
author: Hong Zhang
published in: Beiträge zur Algebra und Geometrie
publication date: 2007-05-14T00:00:00Z
full work available at URL: https://eudml.org/doc/230021
full work available at URL: 24.html
review text: Let \(\gamma\) be a strictly convex, closed planar curve with length \(L\) and bounding a region with area \(A\). The authors prove the following reverse isoperimetric inequality \[ L^2\leq 4\pi(A+| \Tilde A |), \] where \(\Tilde A\) denotes the oriented area of the domain enclosed by the locus of curvature centers of \(\gamma\), and the equality is attained if and only if \(\gamma\) is a circle. The authors study several interesting properties concerning the locus of the centers of curvature and then obtain the proof integrating the square of the radius of curvature and using both Cauchy's and Blaschke's integral formulas.
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 52A38
zbMATH DE Number: 5152757
zbMATH Keywords: strictly convex curves
zbMATH Keywords: Minkowski's support function
zbMATH Keywords: locus of centers of curvature, reverse isoperimetric inequality
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5152757