Counting the interior points of a point configuration
type: Item
instance of: scholarly article
title: Counting the interior points of a point configuration
zbMATH Open document ID: 0953.52007
DOI: 10.1007/PL00009483
author: Victor Reiner
author: Paul Edelman
published in: Discrete \& Computational Geometry
publication date: 2000-01-30T00:00:00Z
review text: Let \(A\) be a collection of points in \(\mathbb{R}^d\) whose affine span is \(\mathbb{R}^d\), and \(\text{int}(A)\) the set of points in \(A\) that lie in the interior of its convex hull \(\text{conv}(A)\). A subset \(K\subseteq A\) is called free if \(\text{conv} (K)\cap A=K\) and the set of vertices of \(\text{conv}(K)\) is \(K\). The authors prove that \[ \bigl|\text{int}(A) \bigr|= (-1)^{d-1} \sum_{K \text{free}} (-1)^{|K|} |K|, \] a recent conjecture of Ahrens, Gordon, and McMahon, by showing that this formula can be interpreted as a sum of Euler characteristics of certain complexes associated with \(A\), and then computing the homology of these complexes. This method extends to other examples of convex geometries.
Mathematics Subject Classification ID: 52B40
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 51A20
zbMATH DE Number: 1394712
zbMATH Keywords: point configuration
zbMATH Keywords: matroid invariants
zbMATH Keywords: convex geometries
reviewed by: Jonathan Schaer
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1394712