An inclusion-exclusion identity for normal cones of polyhedral sets

TL;DR

This paper establishes an inclusion-exclusion identity involving normal cones of polyhedral sets, extending known formulas to unbounded and not necessarily line-free cases, with implications for geometric analysis.

Contribution

It proves a new inclusion-exclusion identity for normal cones of polyhedral sets, generalizing previous results to unbounded and non-line-free sets.

Findings

The identity holds exactly for bounded polyhedral sets.

The identity equals zero for unbounded, line-free sets.

The result extends to non-line-free polyhedral sets.

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.