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AN INCLUSION–EXCLUSION IDENTITY FOR NORMAL CONES OF POLYHEDRAL SETS

Part of: Polytopes and polyhedra General convexity

Published online by Cambridge University Press:  05 February 2018

Daniel Hug  and
Zakhar Kabluchko
Daniel Hug
Affiliation:
Karlsruhe Institute of Technology (KIT), Department of Mathematics, Englerstr. 2, D-76128 Karlsruhe, Germany email daniel.hug@kit.edu
Zakhar Kabluchko
Affiliation:
Institut für Mathematische Statistik, Westfälische Wilhelms-Universität Münster, Orléans–Ring 10, 48149 Münster, Germany email zakhar.kabluchko@uni-muenster.de
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Abstract

For a non-empty polyhedral set $P\subset \mathbb{R}^{d}$, let ${\mathcal{F}}(P)$ denote the set of faces of $P$, and let $N(P,F)$ be the normal cone of $P$ at the non-empty face $F\in {\mathcal{F}}(P)$. We prove the identity

$$\begin{eqnarray}\mathop{\sum }_{F\in {\mathcal{F}}(P)}(-1)^{\operatorname{dim}F}\unicode[STIX]{x1D7D9}_{F-N(P,F)}=\left\{\begin{array}{@{}ll@{}}1\quad & \text{if }P\text{ is bounded},\\ 0\quad & \text{if }P\text{ is unbounded and line-free}.\end{array}\right.\end{eqnarray}$$
Previously, this formula was known to hold everywhere outside some exceptional set of Lebesgue measure $0$ or for polyhedral cones. The case of a not necessarily line-free polyhedral set is also covered by our general theorem.

MSC classification

Primary: 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces) 52B11: $n$-dimensional polytopes
Secondary: 52A55: Spherical and hyperbolic convexity
Type
Research Article
Information
Mathematika , Volume 64 , Issue 1 , 2018 , pp. 124 - 136
Copyright
Copyright © University College London 2018 

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