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Assume that $\\omega_i \\le 1/d$ for every $i\\in[d+1]$. Then there exists a hyperplane $h$ such that each open halfspace $H$ defined by $h$ satisfies $\\mu_i(H) \\le (\\sum_{j=1}^{d+1} \\mu_j(H))/d$ for every $i \\in [d+1]$ and $\\sum_{j=1}^{d+1} \\mu_j(H) \\ge \\min(1/2, 1-d\\omega) "},"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":"1503.06856","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2015-03-23T21:57:59Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"0cc0e941642ae3ff1edaac8bc850a657b0ca889ca26137d624f1bd73b102750d","abstract_canon_sha256":"2d883a25be60d804a3796cdb0578fae0d8c6a0882731589ddbe5fd19bbd2f7ae"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:58:15.333408Z","signature_b64":"JTYu3ToRmvI+KsQ0O+djnZJDL6qzGOpVZYKtb8F52NB9REVTuM5p0aKrkM47kD+h065XDEPqo138d2RabcaBAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"88491f502154834ebcd0a784257cd3fc48a0fe7bfb3852ce633b0ae9087a6bb3","last_reissued_at":"2026-05-17T23:58:15.332980Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:58:15.332980Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The hamburger theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.MG","authors_text":"Jan Kyn\\v{c}l, Mikio Kano","submitted_at":"2015-03-23T21:57:59Z","abstract_excerpt":"We generalize the ham sandwich theorem to $d+1$ measures in $\\mathbb{R}^d$ as follows. Let $\\mu_1,\\mu_2, \\dots, \\mu_{d+1}$ be absolutely continuous finite Borel measures on $\\mathbb{R}^d$. Let $\\omega_i=\\mu_i(\\mathbb{R}^d)$ for $i\\in [d+1]$, $\\omega=\\min\\{\\omega_i; i\\in [d+1]\\}$ and assume that $\\sum_{j=1}^{d+1} \\omega_j=1$. Assume that $\\omega_i \\le 1/d$ for every $i\\in[d+1]$. Then there exists a hyperplane $h$ such that each open halfspace $H$ defined by $h$ satisfies $\\mu_i(H) \\le (\\sum_{j=1}^{d+1} \\mu_j(H))/d$ for every $i \\in [d+1]$ and $\\sum_{j=1}^{d+1} \\mu_j(H) \\ge \\min(1/2, 1-d\\omega) "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.06856","kind":"arxiv","version":4},"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":"1503.06856","created_at":"2026-05-17T23:58:15.333057+00:00"},{"alias_kind":"arxiv_version","alias_value":"1503.06856v4","created_at":"2026-05-17T23:58:15.333057+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.06856","created_at":"2026-05-17T23:58:15.333057+00:00"},{"alias_kind":"pith_short_12","alias_value":"RBER6UBBKSBU","created_at":"2026-05-18T12:29:39.896362+00:00"},{"alias_kind":"pith_short_16","alias_value":"RBER6UBBKSBU5PGQ","created_at":"2026-05-18T12:29:39.896362+00:00"},{"alias_kind":"pith_short_8","alias_value":"RBER6UBB","created_at":"2026-05-18T12:29:39.896362+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/RBER6UBBKSBU5PGQU6CCK7GT7R","json":"https://pith.science/pith/RBER6UBBKSBU5PGQU6CCK7GT7R.json","graph_json":"https://pith.science/api/pith-number/RBER6UBBKSBU5PGQU6CCK7GT7R/graph.json","events_json":"https://pith.science/api/pith-number/RBER6UBBKSBU5PGQU6CCK7GT7R/events.json","paper":"https://pith.science/paper/RBER6UBB"},"agent_actions":{"view_html":"https://pith.science/pith/RBER6UBBKSBU5PGQU6CCK7GT7R","download_json":"https://pith.science/pith/RBER6UBBKSBU5PGQU6CCK7GT7R.json","view_paper":"https://pith.science/paper/RBER6UBB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1503.06856&json=true","fetch_graph":"https://pith.science/api/pith-number/RBER6UBBKSBU5PGQU6CCK7GT7R/graph.json","fetch_events":"https://pith.science/api/pith-number/RBER6UBBKSBU5PGQU6CCK7GT7R/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RBER6UBBKSBU5PGQU6CCK7GT7R/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RBER6UBBKSBU5PGQU6CCK7GT7R/action/storage_attestation","attest_author":"https://pith.science/pith/RBER6UBBKSBU5PGQU6CCK7GT7R/action/author_attestation","sign_citation":"https://pith.science/pith/RBER6UBBKSBU5PGQU6CCK7GT7R/action/citation_signature","submit_replication":"https://pith.science/pith/RBER6UBBKSBU5PGQU6CCK7GT7R/action/replication_record"}},"created_at":"2026-05-17T23:58:15.333057+00:00","updated_at":"2026-05-17T23:58:15.333057+00:00"}