{"paper":{"title":"An inclusion-exclusion identity for normal cones of polyhedral sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Daniel Hug, Zakhar Kabluchko","submitted_at":"2016-12-23T09:40:59Z","abstract_excerpt":"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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.07915","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}