Triangles in convex distance planes
type: Item
instance of: scholarly article
title: Triangles in convex distance planes
zbMATH Open document ID: 1408.52001
published in: Beiträge zur Algebra und Geometrie
publication date: 2018-11-02T00:00:00Z
full work available at URL: http://hdl.handle.net/10138/306738
review text: The author studies triangles in two-dimensional vector spaces with a \textit{convex distance function} (or \textit{gauge}), which is a distance given by the Minkowski functional of a (possibly non-symmetric with respect to the origin) convex body containing the origin as an interior point. Notions of equilateral and acute triangles are extended to this context. The \textit{radius} of a given set \(A\) is defined to be infimum of the radius of the disks containing \(A\), and a disc of minimum radius containing \(A\) is said to be a \textit{minimal enclosing disc}. A triangle is said to be \textit{equilateral} if all of its sides have the same radius, and \textit{weakly acute} if each side has a minimal enclosing disc which does not contain the respective opposite vertex as an interior point. The author proves that every equilateral triangle is weakly acute. Also, the author investigates \textit{central sets} and \textit{bisectors}, which are, roughly speaking, sets of equidistant points of a given set. It is proved that the central set of a weakly acute triangle is non-empty, and that the bisector of an equilateral triangle is also non-empty.
reviewed by: Vitor Balestro
Mathematics Subject Classification ID: 52A10
Mathematics Subject Classification ID: 52A21
zbMATH DE Number: 6971515
zbMATH Keywords: convex distance
zbMATH Keywords: bisector
zbMATH Keywords: central set
zbMATH Keywords: acute
author: Jussi Väisälä
MaRDI profile type: Publication
OpenAlex ID: W2791329376
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Wikidata QID: Q130187910
DOI: 10.1007/S13366-018-0389-3
description: scientific article; zbMATH DE number 6971515