On the self-perimeter of quadrangles for gauges
type: Item
instance of: scholarly article
title: On the self-perimeter of quadrangles for gauges
zbMATH Open document ID: 1222.52001
author: Horst Martini
author: A. I. Shcherba
published in: Beiträge zur Algebra und Geometrie
publication date: 2011-05-16T00:00:00Z
review text: Let \(B\) be a compact convex set of an affine plane \(A^2\) and let \(N\) be a point in \(Int(B)\). A plane \(A^2\) endowed with a gauge metric (possibly non-symmetric) derived from \(B\) and \(N\) is called a Minkowski plane, \(M^2\). The point \(N\) is said to be the origin of \(M^2\), while \(B\) is called the normalizing figure, or the unit ball, of \(M^2\). We thus have a well-defined Minkowski length of a segment and, consequently, a Minkowski length of any curve in \(M^2\). For a non-symmetric metric, this length is sensitive to orientation. If \(P\) is a convex polygon in \(M^2\), let \(l^+(P)\) denote the perimeter of \(P\) measured in the counter-clockwise orientation, while we let \(l^-(P)\) be the perimeter of \(P\) measured clockwise. For \(K\) a compact, convex set in \(M^2\), we define \(L^{\pm}(K)=\sup l^{\pm}(P)\) after all \(P\)'s convex polygons inscribed in \(K\). Various questions regarding sharp bounds on the perimeter of \(B\) itself were posed during the years. Some have been answered. As an example it is known since the 30's (due to Golab) that, if \(B\) is symmetric with respect to \(N\), thus \(L(B)=L^+(B)=L^-(B)\), then \(6 \leq L(B) \leq 8\). There are still several open questions. In particular, Diskant asked about the lower bound of the perimeter of unit balls which are convex \(4\)- and \(5\)-gons. This has been motivated by the result \(\inf L^{\pm}(P_n)=6\), where \(P_n\) is a convex \(n\)-gon, which prompted Makeev to ask whether \(L^{\pm}(P_4)\geq 8\). In view of this, the authors of this paper prove that \[ L^{\pm}(P_4)\geq 3 \cdot (x_0+1) + \frac{1}{x_0} \approx 7.7290, \] where \(x_0\) denotes a real root of the equation \(x^3-x-1=0\).
reviewed by: Alina Stancu
Mathematics Subject Classification ID: 52A10
Mathematics Subject Classification ID: 52A21
Mathematics Subject Classification ID: 52A38
zbMATH DE Number: 5888458
zbMATH Keywords: convex distance functions
zbMATH Keywords: gauges
zbMATH Keywords: Minkowski plane
zbMATH Keywords: normalizing quadrangle
zbMATH Keywords: normed planes
zbMATH Keywords: self-perimeter
zbMATH Keywords: unit circles
MaRDI profile type: Publication
full work available at URL: s13366-011-0018-x
OpenAlex ID: W2055367171
cites work: Q4753788
cites work: Mixed areas and the self-circumference of a plane convex body
cites work: Some properties of the self-circumference of convex sets
cites work: Self-circumference of rotors
cites work: Q5735694
cites work: The perimeter f Minkowski unit discs
cites work: Q3862688
cites work: The geometry of Minkowski spaces -- a survey. I
cites work: Q4449324
cites work: Q4716272
DOI: 10.1007/S13366-011-0018-X
description: scientific article; zbMATH DE number 5888458