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
Article
  • 91 Downloads

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–Schläfli–Poincaré relation and is of independent interest.

Keywords

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

Mathematics Subject Classification

52A22 52B11 60D05 52A05 52B05 

References

  1. 1.
    Affentranger, F.: Generalization of a formula of C. Buchta about the convex hull of random points. Elem. Math. 43(2), 39–45 (1988)MathSciNetMATHGoogle Scholar
  2. 2.
    Affentranger, F.: Remarks on the note: “Generalization of a formula of C. Buchta about the convex hull of random points”. Elem. Math. 43(5), 151–152 (1988)MathSciNetMATHGoogle Scholar
  3. 3.
    Badertscher, E.: An explicit formula about the convex hull of random points. Elem. Math. 44(4), 104–106 (1989)MathSciNetMATHGoogle Scholar
  4. 4.
    Beermann, M., Reitzner, M.: Beyond the Efron–Buchta identities: distributional results for Poisson polytopes. Discrete Comput. Geom. 53(1), 226–244 (2015)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Buchta, C.: On a conjecture of R.E. Miles about the convex hull of random points. Monatsh. Math. 102(2), 91–102 (1986)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Buchta, C.: Distribution-independent properties of the convex hull of random points. J. Theor. Probab. 3(3), 387–393 (1990)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Buchta, C.: An identity relating moments of functionals of convex hulls. Discrete Comput. Geom. 33(1), 125–142 (2005)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cowan, R.: Identities linking volumes of convex hulls. Adv. Appl. Probab. 39(3), 630–644 (2007)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Cowan, R.: Recurrence relationships for the mean number of faces and vertices for random convex hulls. Discrete Comput. Geom. 43(2), 209–220 (2010)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Groemer, H.: Eulersche Charakteristik Projektionen und Quermaßintegrale. Math. Ann. 198(1), 23–56 (1972)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Grünbaum, B.: Convex Polytopes. Graduate Texts in Mathematics, vol. 221, 2nd edn. Springer, New York (2003)CrossRefMATHGoogle Scholar
  12. 12.
    Kabluchko, Z., Vysotsky, V., Zaporozhets, D.: Convex hulls of random walks, hyperplane arrangements, and Weyl chambers (2015). arXiv:1510.04073
  13. 13.
    Nef, W.: Zur Einführung der Eulerschen Charakteristik. Monatsh. Math. 92(1), 41–46 (1981)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory. Encyclopedia of Mathematics and Its Applications, vol. 151, 2nd edn. Cambridge University Press, Cambridge (2014)Google Scholar
  15. 15.
    Schneider, R., Weil, W.: Stochastic and Integral Geometry. Probability and Its Applications. Springer, Berlin (2008)CrossRefMATHGoogle Scholar

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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