A constant bound for geometric permutations of disjoint unit balls
type: Item
instance of: scholarly article
title: A constant bound for geometric permutations of disjoint unit balls
zbMATH Open document ID: 1029.52001
author: Subhash Suri
author: Yunhong Zhou
author: Meir Katchalski
published in: Discrete \& Computational Geometry
publication date: 2003-08-21T00:00:00Z
review text: An oriented line is called a line transversal for a set \(S\) of bodies in the \(d\)-dimensional euclidean space if it intersects every member of \(S\). If the objects in \(S\) are pairwise disjoint convex bodies, then a line transversal of \(S\) induces two linear orderings, which are the orders in which the line meets the members of \(S\). The two orders induced by a line transversal are essentially the same (one is the reverse of the other) and so they are together called a single geometric permutation. In the planar case the maximum number of geometric permutations of a family \(S\) with \(n\) members is known to be \(2n-2\). However for dimension \(d>2\) there is a big gap between the upper and the lower bound. This gap motivated to consider a natural specialized family of convex bodies: unit balls. The authors prove that the maximum number of geometric permutations for a family of \(n\) pairwise disjoint unit balls in the \(d\)-dimensional euclidean space is four, for \(n\) a sufficiently large constant that depends on the dimension \(d\).
Mathematics Subject Classification ID: 52A37
Mathematics Subject Classification ID: 52A35
Mathematics Subject Classification ID: 52C45
zbMATH DE Number: 1969081
zbMATH Keywords: line transversal
zbMATH Keywords: geometric permutation
zbMATH Keywords: combinatorial geometry
MaRDI profile type: Publication
full work available at URL: s00454-002-2828-y
OpenAlex ID: W2024013844
DOI: 10.1007/S00454-002-2828-Y
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description: scientific article; zbMATH DE number 1969081