On antipodal and adjoint pairs of points for two convex bodies
type: Item
instance of: scholarly article
title: On antipodal and adjoint pairs of points for two convex bodies
zbMATH Open document ID: 0841.52002
DOI: 10.1007/BF02574069
published in: Discrete \& Computational Geometry
publication date: 1995-07-02T00:00:00Z
full work available at URL: https://eudml.org/doc/131387
review text: Let \(K_1\), \(K_2\) be two disjoint convex bodies in Euclidean \(d\)-space. A pair of points \(x_1 \in K_1\), \(x_2 \in K_2\) is called strictly adjoint if there exist parallel hyperplanes \(H_1\) through \(x_1\) and \(H_2\) through \(x_2\) both separating \(K_1\) and \(K_2\) such that \(H_i \cap K_i = \{x_i\}\) for \(i = 1,2\). Similarly, \(x_1 \in K_1\) and \(x_2 \in K_2\) are called strictly antipodal if for \(i = 1,2\) there exist hyperplanes \(H_i\) through \(x_i\) such that \(K_1\) and \(K_2\) both lie in the same half-space determined by \(H_i\) and \(H_i \cap K_i = \{x_i\}\). Let \(\overline{p}(K_1,K_2)\) \([\overline{q}(K_1,K_2)]\) denote the number of strictly antipodal [strictly adjoint] pairs of points of \(K_1\) and \(K_2\). The author shows that for any two \(K_1\), \(K_2\) there exists at least one pair of adjoint points, i.e. \(\overline{p} (K_1, K_2) \geq 1\), and at least \(d\) pairs of antipodal points, i.e. \(\overline{q} (K_1, K_2) \geq d\). He further shows: If \(\overline {p}(K_1, K_2')\) is finite for every translate \(K_2'\) of \(K_2\) then \(K_1\) and \(K_2\) are polytopes, and vice versa. The same result is true if \(\overline{p}\) is replaced by \(\overline{q}\). There are some more results in the same spirit.
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52B11
zbMATH DE Number: 764251
zbMATH Keywords: convex bodies
zbMATH Keywords: Euclidean \(d\)-space
zbMATH Keywords: adjoint points
zbMATH Keywords: antipodal points
author: Valeriu Soltan
reviewed by: Bernd Kind
MaRDI profile type: Publication
cites work: Q4774676
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description: scientific article; zbMATH DE number 764251