A characterization of simplices in terms of visibility
type: Item
instance of: scholarly article
title: A characterization of simplices in terms of visibility
zbMATH Open document ID: 0941.52004
author: Horst Martini
author: Valeriu Soltan
published in: Archiv der Mathematik
publication date: 2000-07-31T00:00:00Z
review text: Let \(K\subset {\mathbb R}^d\) be a compact convex set with non-empty interior (convex body). A set \(X\subset {\mathbb R}^d\setminus K\) is said to see the whole boundary of \(K\) if for each boundary point \(z\) of \(K\) there is some \(x\in X\) such that \(K\) has empty intersection with the open segment \((x,z)\). Extending an observation of \textit{V. Soltan} [see Theorem 1 in Stud. Sci. Math. Hung. 28, No. 3-4, 473-483 (1993; Zbl 0817.52013)], the authors prove that for any point \(x_1\in {\mathbb R}^d\setminus K\) there exist \(d\) points \(x_2,\ldots,x_{d+1}\in{\mathbb R}^d\setminus K\) such that \(\{x_1,\ldots,x_{d+1}\}\) sees the whole boundary of \(K\). The main result of the paper states that a convex body \(K\subset {\mathbb R}^d\) is a simplex if and only if for any point \(x_1\in {\mathbb R}^d\setminus K\) there exists a point \(x_2\in {\mathbb R}^d\setminus K\) such that \(\{x_1,x_2\}\) sees the whole boundary of \(K\).
reviewed by: Daniel Hug
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52A37
Mathematics Subject Classification ID: 52A30
zbMATH DE Number: 1333335
zbMATH Keywords: convex body
zbMATH Keywords: simplex
zbMATH Keywords: visibility
zbMATH Keywords: positive basis
MaRDI profile type: Publication
full work available at URL: s000130050356
OpenAlex ID: W2090990457
DOI: 10.1007/S000130050356
description: scientific article; zbMATH DE number 1333335