On the connected components of the space of line transversals to a family of convex sets
type: Item
instance of: scholarly article
title: On the connected components of the space of line transversals to a family of convex sets
zbMATH Open document ID: 0824.52004
DOI: 10.1007/BF02574057
published in: Discrete \& Computational Geometry
publication date: 1995-07-02T00:00:00Z
full work available at URL: https://eudml.org/doc/131375
review text: Let \(A\) be a family of convex sets in \(\mathbb{R}^ d\). A \(k\)-transversal for \(A\) is a \(k\)-flat (= affine subspace of dimension \(k\)) that intersects every member of \(A\). This paper deals with the connected components of the space \(A^*\) consisting of all \(k\)-transversals of \(A\) (considered as a subspace of the affine Grassmannian of all \(k\)-flats in \(\mathbb{R}^ d\)). If \(F\) is a convex set of \(k\)-flats in \(\mathbb{R}^ d\), then \(F= A^*\) for some family of convex point sets in \(\mathbb{R}^ d\); \(F\) is then said to be represented by \(A\). A convex set \(F\) of \(k\)-flats need not be connected for \(k>0\), and a connected component of \(F\) need not be convex, as the authors show by exhibiting an example of a convex family of lines in \(\mathbb{R}^ 3\) that has a nonconvex connected component. If, however, a convex set \(F\) of \(k\)-flats in \(\mathbb{R}^ d\) is represented by some finite family of suitably separated compact convex point sets, then the authors conjecture that the connected components of \(F\) will themselves be convex. As the main result of this article, this conjecture is proved for the case \(d=3\), \(k=1\): Let \(F\) be a convex set of lines in \(\mathbb{R}^ 3\) represented by a finite family of pairwise disjoint compact convex point sets; then each connected component of \(F\) can itself be represented as the space of transversals to some finite family of pairwise disjoint compact convex sets.
Mathematics Subject Classification ID: 52A15
zbMATH DE Number: 764239
zbMATH Keywords: family of convex sets
zbMATH Keywords: line transversals
zbMATH Keywords: \(k\)-transversals
author: Jacob E. Goodman
author: Richard Pollack
author: Rephael Wenger
MaRDI profile type: Publication
cites work: Q4274009
cites work: Foundations of a theory of convexity on affine Grassmann manifolds
cites work: Q3138749
cites work: Geometric permutations for convex sets
description: scientific article; zbMATH DE number 764239