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Our main result states that for a compact domain $M \\subset \\mathbb{R}^d$ with piecewise $C^1$ boundary and bounded $N \\subset \\mathbb{R}^d$, $D_N(M)=D_{\\text{conv}(N)}(M)$ and $D_N(M)=\\int_{\\text{bd }M} h_N(u_M(x)) \\, d \\mathcal{H}^{d-1}(x)$."},"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":"1607.07802","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-07-26T16:51:47Z","cross_cats_sorted":[],"title_canon_sha256":"25b770214d59ec1816d4dc0c4fa660b5bdfc638572e7747226538129321fd766","abstract_canon_sha256":"01a3028692c360798f20a20733e55f8735425dea3c26cb71101bff6d78bdfbd4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:10:25.149217Z","signature_b64":"O1ZYzNCF+CjBvEubFgcmPNKPNvJQQjorOT+MWuTbvSn2TSyGritSUiUPh+WrtgOTWCfHgc8EnAIXR6273t0KDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a5c4ca7a2f67c6d0fcdea83d9ac23d3bcd5aab136601a16dce82beed3519231a","last_reissued_at":"2026-05-18T01:10:25.148640Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:10:25.148640Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Extension of the first mixed volume to nonconvex sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Emmanuel Tsukerman","submitted_at":"2016-07-26T16:51:47Z","abstract_excerpt":"We study the first mixed volume for nonconvex sets and apply the results to limits of discrete isoperimetric problems. Let $ M,N \\subset \\mathbb{R}^d$. Define $D_N (M)=\\lim_{\\epsilon \\downarrow 0} \\frac{|M+\\epsilon N|-|M|}{\\epsilon}$ whenever the limit exists. Our main result states that for a compact domain $M \\subset \\mathbb{R}^d$ with piecewise $C^1$ boundary and bounded $N \\subset \\mathbb{R}^d$, $D_N(M)=D_{\\text{conv}(N)}(M)$ and $D_N(M)=\\int_{\\text{bd }M} h_N(u_M(x)) \\, d \\mathcal{H}^{d-1}(x)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.07802","kind":"arxiv","version":1},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1607.07802","created_at":"2026-05-18T01:10:25.148709+00:00"},{"alias_kind":"arxiv_version","alias_value":"1607.07802v1","created_at":"2026-05-18T01:10:25.148709+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.07802","created_at":"2026-05-18T01:10:25.148709+00:00"},{"alias_kind":"pith_short_12","alias_value":"UXCMU6RPM7DN","created_at":"2026-05-18T12:30:46.583412+00:00"},{"alias_kind":"pith_short_16","alias_value":"UXCMU6RPM7DNB7G6","created_at":"2026-05-18T12:30:46.583412+00:00"},{"alias_kind":"pith_short_8","alias_value":"UXCMU6RP","created_at":"2026-05-18T12:30:46.583412+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/UXCMU6RPM7DNB7G6VA6ZVQR5HP","json":"https://pith.science/pith/UXCMU6RPM7DNB7G6VA6ZVQR5HP.json","graph_json":"https://pith.science/api/pith-number/UXCMU6RPM7DNB7G6VA6ZVQR5HP/graph.json","events_json":"https://pith.science/api/pith-number/UXCMU6RPM7DNB7G6VA6ZVQR5HP/events.json","paper":"https://pith.science/paper/UXCMU6RP"},"agent_actions":{"view_html":"https://pith.science/pith/UXCMU6RPM7DNB7G6VA6ZVQR5HP","download_json":"https://pith.science/pith/UXCMU6RPM7DNB7G6VA6ZVQR5HP.json","view_paper":"https://pith.science/paper/UXCMU6RP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1607.07802&json=true","fetch_graph":"https://pith.science/api/pith-number/UXCMU6RPM7DNB7G6VA6ZVQR5HP/graph.json","fetch_events":"https://pith.science/api/pith-number/UXCMU6RPM7DNB7G6VA6ZVQR5HP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UXCMU6RPM7DNB7G6VA6ZVQR5HP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UXCMU6RPM7DNB7G6VA6ZVQR5HP/action/storage_attestation","attest_author":"https://pith.science/pith/UXCMU6RPM7DNB7G6VA6ZVQR5HP/action/author_attestation","sign_citation":"https://pith.science/pith/UXCMU6RPM7DNB7G6VA6ZVQR5HP/action/citation_signature","submit_replication":"https://pith.science/pith/UXCMU6RPM7DNB7G6VA6ZVQR5HP/action/replication_record"}},"created_at":"2026-05-18T01:10:25.148709+00:00","updated_at":"2026-05-18T01:10:25.148709+00:00"}