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arXiv · 1612.07915

An inclusion-exclusion identity for normal cones of polyhedral sets

Daniel Hug·Zakhar Kabluchko
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Abstract

For a nonempty 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 nonempty face $F\in\mathcal F(P)$. We prove that the function $\sum_{F\in\mathcal F(P)}(-1)^{\text{dim} F} 1_{F-N(P,F)}$ equals $1$ if $P$ is bounded, or $0$ if $P$ is unbounded and line-free. 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.

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Daniel Hug, Zakhar Kabluchko. 2016-12-23. An inclusion-exclusion identity for normal cones of polyhedral sets. https://doi.org/10.1112/s0025579317000390

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