The \(\epsilon\)-positive center figure
type: Item
instance of: scholarly article
title: The \(\epsilon\)-positive center figure
zbMATH Open document ID: 0877.52004
DOI: 10.1016/0893-9659(96)00096-1
published in: Applied Mathematics Letters
publication date: 1997-02-13T00:00:00Z
review text: The area of a circle is usually given as \(\pi R^2\), but it is possibly more illuminating to consider it as \(RC/2\). Any other convex curve does not have a unique radius, but the support function \(H(K,u):S^1\to\mathbb{R}\) with respect to a point \(p\) is one possible generalization. Gage showed, in 1990, that every convex curve \(K\) with length \(L(K)\) and area \(A(K)\) contains at least one point \(p\) such that \(H(K,u) L(K) -A(K)- \pi H(K,u)^2 \geq 0\) for all \(u\in S^1\). Such a point is called a positive center of \(K\). In general such points are not unique. The current author extends the definition, defining an \(\varepsilon\)-positive center to be a point such that \(H(K,u)L(K)- A(K)- \pi H(K,u)^2 \geq \varepsilon\) for all \(u\in S^1\) and some \(\varepsilon>0\). Two examples are shown. Two conjectures are also given: that at least one \(\varepsilon\)-positive center always exists for some \(\varepsilon>0\) whenever the positive center is not unique, and that the set of such points is always convex. (Let \(H(K,u, p)\) be the support function of \(K\), in direction \(u\), with the point \(p\) taken as origin. Then, if \(p= tp_1+ (1-t)p_2\), \(H(K,u,p) =tH(K,u,p_1)+ (1-t)H(K,u,p_2)\). Thus, \(H(K,u,p) L(K)-A(K) -\pi H(K,u,p)^2\) is convex upwards as a function of \(t\), and strictly so when \(u\) is not normal to \(p_1- p_2\). This proves the second conjecture, and appears to make some headway on the first).
Mathematics Subject Classification ID: 52A10
Mathematics Subject Classification ID: 52A41
Mathematics Subject Classification ID: 51M15
zbMATH DE Number: 978094
zbMATH Keywords: \(\varepsilon\)-positive center
zbMATH Keywords: convex curve
zbMATH Keywords: length
zbMATH Keywords: area
author: Mark J. Kaiser
reviewed by: Robert Dawson
MaRDI profile type: Publication
full work available at URL: 0893-9659(96)00096-1
OpenAlex ID: W2059299913
cites work: Positive Centers and the Bonnesen Inequality
cites work: Duality of convex bodies
cites work: Q3764803
description: scientific article; zbMATH DE number 978094