Helly-type theorems for spheres
type: Item
instance of: scholarly article
title: Helly-type theorems for spheres
zbMATH Open document ID: 0671.52009
DOI: 10.1007/BF02187730
author: H. Maehara
published in: Discrete \& Computational Geometry
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/131079
review text: A sphere S of radius r in Euclidean n-space \(E^ n\) consists of all points at distance r from some fixed point called the center of S. Let F be a family of spheres in \(E^ n\). The author proves the following theorems: 1) If the centers of the spheres in F are on a flat of dimension m and every subfamily consisting of at most \(m+2\) spheres in F has nonempty intersection, then F has nonempty intersection. 2) If F has nonempty intersection and the centers of the spheres in F all lie on a flat of dimension m, then there exist \(m+1\) of fewer spheres in F whose intersection is equal to the intersection of all spheres of F. 3) If F has at least \(n+3\) different spheres and any \(n+1\) spheres in F have a common point, then F has nonempty intersection.
Mathematics Subject Classification ID: 52A35
zbMATH DE Number: 4098187
zbMATH Keywords: Helly property
zbMATH Keywords: spheres in \(E^ n\)
reviewed by: Radu Precup
MaRDI profile type: Publication
cites work: Q5341481
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description: scientific article; zbMATH DE number 4098187