Inclusion–Exclusion Principles for Convex Hulls and the Euler Relation

Abstract

Consider n points \(X_1,\ldots ,X_n\) in \(\mathbb {R}^d\) and denote their convex hull by \({\Pi }\). We prove a number of inclusion–exclusion identities for the system of convex hulls \({\Pi }_I:=\mathrm{conv}(X_i: i\in I)\), where I ranges over all subsets of \(\{1,\ldots ,n\}\). For instance, denoting by \(c_k(X)\) the number of k-element subcollections of \((X_1,\ldots ,X_n)\) whose convex hull contains a point \(X\in \mathbb {R}^d\), we prove that $$\begin{aligned} c_1(X)-c_2(X)+c_3(X)-\cdots + (-1)^{n-1} c_n(X) = (-1)^{\dim {\Pi }} \end{aligned}$$ for allX in the relative interior of \({\Pi }\). This confirms a conjecture of Cowan (Adv Appl Probab 39(3):630–644, 2007) who proved the above formula for almost allX. We establish similar results for the number of polytopes \({\Pi }_J\) containing a given polytope \({\Pi }_I\) as an r-dimensional face, thus proving another conjecture of Cowan (Discrete Comput Geom 43(2):209–220, 2010). As a consequence, we derive inclusion–exclusion identities for the intrinsic volumes and the face numbers of the polytopes \({\Pi }_I\). The main tool in our proofs is a formula for the alternating sum of the face numbers of a convex polytope intersected by an affine subspace. This formula generalizes the classical Euler–Schlafli–Poincare relation and is of independent interest.