Inclusion-Exclusion Principles for Convex Hulls and the Euler Relation

Abstract. Consider n points X1, . . . ,Xn in R and denote their convex hull by Π. We prove a number of inclusion-exclusion identities for the system of convex hulls ΠI := conv(Xi : i ∈ I), where I ranges over all subsets of {1, . . . , n}. For instance, denoting by ck(X) the number of k-element subcollections of (X1, . . . ,Xn) whose convex hull contains a point X ∈ R, we prove that c1(X) − c2(X) + c3(X) − . . .+ (−1) n−1cn(X) = (−1) dimΠ