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Here, $\\theta^+ := \\lbrace x\\, : \\, \\langle x,\\theta\\rangle \\geq 0 \\rbrace$. As a consequence of this result, we get the following "},"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":"1711.00998","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2017-11-03T02:18:32Z","cross_cats_sorted":[],"title_canon_sha256":"1025273e9b43889520c652552b5d5ed6d02e50234209bb0d6fc2edc7105ad81a","abstract_canon_sha256":"0087b320e4926a7bf5ab27ebda864b1bcab40d8643af51f1cfccfbf294c3121d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:31:24.322252Z","signature_b64":"+YAmVSG9s4RVzF8UAMIaYlS9CMBbF/w5/tMN11uAtF68dJSZWvxkFIKWXFZJDaryzc4zvVKAeYkJ0VfriLYkAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"94061961cbe6a700f4d3134859a0b6cc04764b5ee214c20e260a9225bf621748","last_reissued_at":"2026-05-18T00:31:24.321753Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:31:24.321753Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Gr\\\"unbaum's inequality for sections","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Matthew Stephen, Ning Zhang, Sergii Myroshnychenko","submitted_at":"2017-11-03T02:18:32Z","abstract_excerpt":"We show \\begin{align*} \\frac{ \\int_{E \\cap \\theta^+} f(x) dx }{ \\int_E f(x) dx }\n  \\geq \\left(\\frac{k \\gamma+1}{(n+1) \\gamma+1}\\right)^{\\frac{k \\gamma+1}{\\gamma}} \\end{align*} for all $k$-dimensional subspaces $E\\subset\\mathbb{R}^n$, $\\theta\\in E\\cap S^{n-1}$, and all $\\gamma$-concave functions $f:\\mathbb{R}^n\\rightarrow [0,\\infty)$ with $\\gamma >0$, $0< \\int_{\\mathbb{R}^n} f(x)\\, dx <\\infty$, and $\\int_{\\mathbb{R}^n} x f(x)\\, dx$ at the origin $o\\in\\mathbb{R}^n$. Here, $\\theta^+ := \\lbrace x\\, : \\, \\langle x,\\theta\\rangle \\geq 0 \\rbrace$. As a consequence of this result, we get the following "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.00998","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":"1711.00998","created_at":"2026-05-18T00:31:24.321816+00:00"},{"alias_kind":"arxiv_version","alias_value":"1711.00998v1","created_at":"2026-05-18T00:31:24.321816+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.00998","created_at":"2026-05-18T00:31:24.321816+00:00"},{"alias_kind":"pith_short_12","alias_value":"SQDBSYOL42TQ","created_at":"2026-05-18T12:31:43.269735+00:00"},{"alias_kind":"pith_short_16","alias_value":"SQDBSYOL42TQB5GT","created_at":"2026-05-18T12:31:43.269735+00:00"},{"alias_kind":"pith_short_8","alias_value":"SQDBSYOL","created_at":"2026-05-18T12:31:43.269735+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/SQDBSYOL42TQB5GTCNEFTIFWZQ","json":"https://pith.science/pith/SQDBSYOL42TQB5GTCNEFTIFWZQ.json","graph_json":"https://pith.science/api/pith-number/SQDBSYOL42TQB5GTCNEFTIFWZQ/graph.json","events_json":"https://pith.science/api/pith-number/SQDBSYOL42TQB5GTCNEFTIFWZQ/events.json","paper":"https://pith.science/paper/SQDBSYOL"},"agent_actions":{"view_html":"https://pith.science/pith/SQDBSYOL42TQB5GTCNEFTIFWZQ","download_json":"https://pith.science/pith/SQDBSYOL42TQB5GTCNEFTIFWZQ.json","view_paper":"https://pith.science/paper/SQDBSYOL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1711.00998&json=true","fetch_graph":"https://pith.science/api/pith-number/SQDBSYOL42TQB5GTCNEFTIFWZQ/graph.json","fetch_events":"https://pith.science/api/pith-number/SQDBSYOL42TQB5GTCNEFTIFWZQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SQDBSYOL42TQB5GTCNEFTIFWZQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SQDBSYOL42TQB5GTCNEFTIFWZQ/action/storage_attestation","attest_author":"https://pith.science/pith/SQDBSYOL42TQB5GTCNEFTIFWZQ/action/author_attestation","sign_citation":"https://pith.science/pith/SQDBSYOL42TQB5GTCNEFTIFWZQ/action/citation_signature","submit_replication":"https://pith.science/pith/SQDBSYOL42TQB5GTCNEFTIFWZQ/action/replication_record"}},"created_at":"2026-05-18T00:31:24.321816+00:00","updated_at":"2026-05-18T00:31:24.321816+00:00"}