Discrete & Computational Geometry

, Volume 58, Issue 2, pp 417–434 | Cite as

Inclusion–Exclusion Principles for Convex Hulls and the Euler Relation

  • Zakhar Kabluchko
  • Günter Last
  • Dmitry Zaporozhets


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–Schläfli–Poincaré relation and is of independent interest.


Convex hulls Inclusion–exclusion principle Cowan’s formula Euler characteristic Euler relation Polytopes Faces Intrinsic volumes 

Mathematics Subject Classification

52A22 52B11 60D05 52A05 52B05 


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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Universität Münster, Institut für Mathematische StochastikMünsterGermany
  2. 2.Karlsruher Institut für TechnologieKarlsruheGermany
  3. 3.St. Petersburg Department of Steklov Institute of MathematicsSaint PetersburgRussia

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