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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."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1612.07915","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-12-23T09:40:59Z","cross_cats_sorted":[],"title_canon_sha256":"ae49f81fdedde0cf5cd97f5c9d3e9aec7d1f0626908307cde2a820d56ff05ab9","abstract_canon_sha256":"ae161f52b37e94a382e1bb8f31db06e3dcd2a52ead1fd53659e0269e8806cb05"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:23:40.361888Z","signature_b64":"ngIEN+Ab2iBCPuDOdZIHcnc1YjsM7c0f02DrG8+INeG46ATvYOqA8PZv4R30El2MSND+RDFPIFzp7qu15CdiBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"95808dc1c30a45bc7b4d6e77faf8c994c99397ee9143f176647c6ccee802807b","last_reissued_at":"2026-05-18T00:23:40.361116Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:23:40.361116Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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