Convexity and the beta invariant
type: Item
instance of: scholarly article
title: Convexity and the beta invariant
zbMATH Open document ID: 0966.52006
DOI: 10.1007/PL00009469
author: Yanyan Li
published in: Discrete \& Computational Geometry
publication date: 2001-08-06T00:00:00Z
review text: Let \(S\) be a finite nonempty subset of \(\mathbb R^n\) with free convex set \(\mathcal C\). A convex set \(\mathcal C\) is called free if every subset of \(\mathcal C\) is also convex. Then the beta invariant of \(S\) is defined as \[ \beta (S) = \sum_{K\in{\mathcal C}} (-1)^{|K|-1}|K|. \] The boundary \(B(S)\) of \(S\) is the collection of points of \(S\) which are on the boundary of the convex hull of \(S\). The interior \(\text{int} (S)\) of \(S\) is the complement of the boundary: \(\text{int} (S)=S \setminus B(S)\). The main result of the paper is a geometric interpretation for \(\beta (S)\) in the plane: Let \(S\) be a finite subset of \(\mathbb R^2\) which does not lie on a line. Then \(\beta (S)=|\text{int} (S)|\).
reviewed by: Arnfried Kemnitz
Mathematics Subject Classification ID: 52A37
Mathematics Subject Classification ID: 52A20
zbMATH DE Number: 1367686
zbMATH Keywords: convexity
zbMATH Keywords: beta invariant
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1367686