Helly-type theorems on the homology of the space of transversals
type: Item
instance of: scholarly article
title: Helly-type theorems on the homology of the space of transversals
zbMATH Open document ID: 1021.52005
published in: Discrete \& Computational Geometry
publication date: 2002-05-30T00:00:00Z
review text: An \(n\)-dimensional flat \(P\) is a \(n\)-transversal of a family \({\mathcal C}= \{K_i\}_{i=0}^d\) of convex bodies in \({R}^{n+k}\) if \(P\) intersects \(K_i\) for each \(i=0,\ldots, d\). Let \(AG_n(R^{n+k})\), respectively \(G_n(R^{n+k})\), be the Grassmannian of all affine, respectively linear, subspaces in \(R^{n+k}\). Let \(T_n({\mathcal C})\subset AG_n(R^{n+k})\) be the space of all \(n\)-transversals of \({\mathcal C}\). The authors introduce a homological measure of complexity \(\mu(X)\) for subsets \(X\subset AG_n(R^{n+k})\subset G_{n+1}(R^{n+k+1})\) and use it to formulate Helly-type results in the spirit of classical results of Horn and Klee. Informally speaking \(\mu(X)\geq r\) if \(X\) contains a ``translate'' of the fundamental cycle of \(G_n(R^{n+r})\). The main result says that if \(1\leq m\leq n\) then \(\mu(T_n({\mathcal C}))\geq k\) if and only if \(\mu(T_m({\mathcal C}')\geq k\) for each \({\mathcal C}'\subset {\mathcal C}\) of size \(k+2\). For example if \(m,k=1\) and \(n=2\), the result says that each three convex sets in \({\mathcal C}\) admit a non-contractible loop of transversal lines if and only if \({\mathcal C}\) has transversal planes orthogonal to every direction prescribed in advance.
Mathematics Subject Classification ID: 52A35
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 51M35
zbMATH DE Number: 1747948
zbMATH Keywords: Helly-type results
zbMATH Keywords: geometric transversals
author: Javier Bracho
author: L. Montejano
reviewed by: Rade Živaljević
MaRDI profile type: Publication
full work available at URL: s00454-001-0076-1
OpenAlex ID: W2149920101
DOI: 10.1007/S00454-001-0076-1
description: scientific article; zbMATH DE number 1747948