Inclusion-exclusion principles for convex hulls and the Euler relation

TL;DR

This paper establishes new inclusion-exclusion identities for convex hulls and related polytopes, confirming conjectures by R. Cowan and generalizing classical geometric relations, with applications to intrinsic volumes and face numbers.

Contribution

It proves conjectured inclusion-exclusion formulas for convex hulls and polytopes, extending classical geometric relations and providing new tools for polytope analysis.

Findings

Proved inclusion-exclusion identities for convex hulls containing a point.

Confirmed conjectures of R. Cowan regarding polytopes and convex hulls.

Derived identities for intrinsic volumes and face numbers of polytopes.

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 $\Pi_I:=conv(X_i\colon i\in I)$, 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 $$ c_1(X)-c_2(X)+c_3(X)-\ldots + (-1)^{n-1} c_n(X) = (-1)^{\dim \Pi} $$ for all $X$ in the relative interior of $\Pi$. 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 $\Pi_J$ containing a given polytope $\Pi_I$ 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 $\Pi_I$. 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--Schl\"afli--Poincar\'e relation and is of independent interest.