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Inclusion-exclusion principles for convex hulls and the Euler relation

Consider $n$ points $X_1,\ldots,X_n$ in $\mathbb R^d$ and denote their convex hull by $Π$. We prove a number of inclusion-exclusion identities for the system of convex hulls $Π_I:=conv(X_i\colon i\in I)$, 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 $$ c_1(X)-c_2(X)+c_3(X)-\ldots + (-1)^{n-1} c_n(X) = (-1)^{\dim Π} $$ for all $X$ in the relative interior of $Π$. This confirms a conjecture of R. Cowan [Adv. Appl. Probab., 39(3):630--644, 2007] who proved the above formula for almost all $X$. We establish similar results for the number of polytopes $Π_J$ containing a given polytope $Π_I$ as an $r$-dimensional face, thus proving another conjecture of R. 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 $Π_I$. 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--Schläfli--Poincaré relation and is of independent interest.