Bisectors in Minkowski 3-space
type: Item
instance of: scholarly article
title: Bisectors in Minkowski 3-space
zbMATH Open document ID: 1054.52003
published in: Beiträge zur Algebra und Geometrie
publication date: 2004-09-01T00:00:00Z
full work available at URL: https://eudml.org/doc/124126
full work available at URL: 19.html
review text: A 0-symmetric, bounded, convex body \(K\) in Euclidean \(n\)-space \(E^n\) defines a norm whose unit ball is \(K\). It was shown by the author [Acta Math. Hung. 89, 233--246 (2000; Zbl 0973.52001)] that if \(K\) is strictly convex then every bisector \(H_{\mathbf x}\), that is, the collection of all points that are equidistant from 0 and \(x\), is a topological hyperplane. The converse is not true however. In the paper under review the author conjectures a characterisation of topological hyperplanes in terms of shadow boundaries: The bisectors are topological hyperplanes if and only if the corresponding shadow boundaries are \((n-2)\)-dimensional topological spheres. The author proves this conjecture for the 3-dimensional case. In section 2 the author examines the topological properties of shadow boundaries, providing two concrete examples that show that they need neither be arcwise connected nor manifolds. He also introduces general parameter spheres as a tool for a potential proof of his conjecture. The shadow boundary \(S(K,{\mathbf x})\) of \(K\) in direction \({\mathbf x}\) in the unit sphere \(S^{n-1}\) of \(E^n\) consists of all points \(P\) in the boundary \(bdK\) of \(K\) such that the line through \(P\) in direction \({\mathbf x}\) does not meet the interior of \(K\). The general parameter sphere of \(bdK\) in direction \({\mathbf x}\) is \(\gamma_\lambda(K,{\mathbf x})=\frac{1}{\lambda}[bd(\lambda K)\cap bd(\lambda K+{\mathbf x})]\) where \(\lambda\geq \lambda_0\) and \(\lambda_0\) is the smallest value \(t\) for which \(tK\) and \(tK+{\mathbf x}\) intersect. Each bisector \(H_{\mathbf x}\) is the union over all \(\gamma_{\lambda}(K,{\mathbf x})\) for \(\lambda\geq \lambda_0\). The author shows that the general parameter spheres provide a natural parametrization for the points in the positive part of \(bdK\) minus \(\gamma_{\lambda_0}(K,{\mathbf x})\) by a point of \(S^{n-2}\) and a parameter \(\lambda >\lambda_0\). Furthermore, \(S(K,{\mathbf x})\) is th…
reviewed by: Günter F. Steinke
Mathematics Subject Classification ID: 52A21
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 2096243
zbMATH Keywords: Minkowski normed space
zbMATH Keywords: shadow boundary
zbMATH Keywords: convex set
zbMATH Keywords: strict convexity
author: Ákos G. Horváth
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2096243