Rich cells in an arrangement of hyperplanes
type: Item
instance of: scholarly article
title: Rich cells in an arrangement of hyperplanes
zbMATH Open document ID: 0831.52007
DOI: 10.1016/0024-3795(95)00244-L
author: H. Bunting
author: János Pach
author: Imre Bárány
author: D. G. Larman
published in: Linear Algebra and its Applications
publication date: 1996-02-12T00:00:00Z
review text: Let \(A\) denote an arrangement of hyperplanes in \(\mathbb{R}^d\). A \(d\)- dimensional cell of \(A\) is said to be rich if its boundary contains a nondegenerate (i.e., a \((d - 1)\)-dimensional) piece of each of the \(n\) hyperplanes of \(A\). The authors derive bounds on \(f(d,n)\), which is the maximum number of rich cells over all such arrangements (for \(d = 2\), even the precise number is found), and they prove that for \(n \geq d \geq 3\) the relation \[ f(d,n) = {n \choose d - 2} + O(n^{d - 3}) \] holds. Furthermore, they prove a theorem on the ``Carathéodory number'' for lines in the plane: a planar arrangement \(A\) has a rich cell provided each five of the \(n \geq 6\) lines of \(A\) form a partial arrangement containing itself a rich cell.
reviewed by: Horst Martini
Mathematics Subject Classification ID: 52C35
zbMATH DE Number: 803761
zbMATH Keywords: Carathéodory's theorem
zbMATH Keywords: convex position
zbMATH Keywords: arrangement of hyperplanes
MaRDI profile type: Publication
cites work: Combinatorial complexity bounds for arrangements of curves and spheres
cites work: Q5556250
cites work: Foundations of a theory of convexity on affine Grassmann manifolds
cites work: Q5547252
cites work: On Sets Projectively Equivalent to the Vertices of a Convex Polytope
cites work: Hamilton Paths in Tournaments and a Problem of McMullen on Projective Transformations in R d
cites work: Extremal problems in discrete geometry
description: scientific article; zbMATH DE number 803761