Convexity and the beta invariant
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 \(\beta (S)=|\text{int} (S)|\).
scientific article; zbMATH DE number 1367686