{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:SQDBSYOL42TQB5GTCNEFTIFWZQ","short_pith_number":"pith:SQDBSYOL","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"},"canonical_sha256":"94061961cbe6a700f4d3134859a0b6cc04764b5ee214c20e260a9225bf621748","source":{"kind":"arxiv","id":"1711.00998","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1711.00998","created_at":"2026-05-18T00:31:24Z"},{"alias_kind":"arxiv_version","alias_value":"1711.00998v1","created_at":"2026-05-18T00:31:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.00998","created_at":"2026-05-18T00:31:24Z"},{"alias_kind":"pith_short_12","alias_value":"SQDBSYOL42TQ","created_at":"2026-05-18T12:31:43Z"},{"alias_kind":"pith_short_16","alias_value":"SQDBSYOL42TQB5GT","created_at":"2026-05-18T12:31:43Z"},{"alias_kind":"pith_short_8","alias_value":"SQDBSYOL","created_at":"2026-05-18T12:31:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:SQDBSYOL42TQB5GTCNEFTIFWZQ","target":"record","payload":{"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"},"canonical_sha256":"94061961cbe6a700f4d3134859a0b6cc04764b5ee214c20e260a9225bf621748","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"},"source_kind":"arxiv","source_id":"1711.00998","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:31:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vdAWDiIzz6Fjqv7gfOCGJ/x+XChARdg8jK+2wLFdseBY7XXJEM7bYCz+kCjfsFag4XrkhwB3FTPuxgh1TducBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T13:00:38.388690Z"},"content_sha256":"5d3e0869931793c26fd2d649d5eaba698c9ba176a2e81a2cebb69b1a41ec0d57","schema_version":"1.0","event_id":"sha256:5d3e0869931793c26fd2d649d5eaba698c9ba176a2e81a2cebb69b1a41ec0d57"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:SQDBSYOL42TQB5GTCNEFTIFWZQ","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:31:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4ibUgi28MHIM8CPD/OtoNTUrN5W5hvtK9a6LSe/fWULr84wmYxY94GWC0vVE9J8kXt9dkWB08CQyES4wdrYMDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T13:00:38.389421Z"},"content_sha256":"e485bb0670b378835b389f3dd241a646edd92522bf925477c701f3c2baa252f8","schema_version":"1.0","event_id":"sha256:e485bb0670b378835b389f3dd241a646edd92522bf925477c701f3c2baa252f8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SQDBSYOL42TQB5GTCNEFTIFWZQ/bundle.json","state_url":"https://pith.science/pith/SQDBSYOL42TQB5GTCNEFTIFWZQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SQDBSYOL42TQB5GTCNEFTIFWZQ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-27T13:00:38Z","links":{"resolver":"https://pith.science/pith/SQDBSYOL42TQB5GTCNEFTIFWZQ","bundle":"https://pith.science/pith/SQDBSYOL42TQB5GTCNEFTIFWZQ/bundle.json","state":"https://pith.science/pith/SQDBSYOL42TQB5GTCNEFTIFWZQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SQDBSYOL42TQB5GTCNEFTIFWZQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:SQDBSYOL42TQB5GTCNEFTIFWZQ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"0087b320e4926a7bf5ab27ebda864b1bcab40d8643af51f1cfccfbf294c3121d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2017-11-03T02:18:32Z","title_canon_sha256":"1025273e9b43889520c652552b5d5ed6d02e50234209bb0d6fc2edc7105ad81a"},"schema_version":"1.0","source":{"id":"1711.00998","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1711.00998","created_at":"2026-05-18T00:31:24Z"},{"alias_kind":"arxiv_version","alias_value":"1711.00998v1","created_at":"2026-05-18T00:31:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.00998","created_at":"2026-05-18T00:31:24Z"},{"alias_kind":"pith_short_12","alias_value":"SQDBSYOL42TQ","created_at":"2026-05-18T12:31:43Z"},{"alias_kind":"pith_short_16","alias_value":"SQDBSYOL42TQB5GT","created_at":"2026-05-18T12:31:43Z"},{"alias_kind":"pith_short_8","alias_value":"SQDBSYOL","created_at":"2026-05-18T12:31:43Z"}],"graph_snapshots":[{"event_id":"sha256:e485bb0670b378835b389f3dd241a646edd92522bf925477c701f3c2baa252f8","target":"graph","created_at":"2026-05-18T00:31:24Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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 ","authors_text":"Matthew Stephen, Ning Zhang, Sergii Myroshnychenko","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2017-11-03T02:18:32Z","title":"Gr\\\"unbaum's inequality for sections"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.00998","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:5d3e0869931793c26fd2d649d5eaba698c9ba176a2e81a2cebb69b1a41ec0d57","target":"record","created_at":"2026-05-18T00:31:24Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"0087b320e4926a7bf5ab27ebda864b1bcab40d8643af51f1cfccfbf294c3121d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2017-11-03T02:18:32Z","title_canon_sha256":"1025273e9b43889520c652552b5d5ed6d02e50234209bb0d6fc2edc7105ad81a"},"schema_version":"1.0","source":{"id":"1711.00998","kind":"arxiv","version":1}},"canonical_sha256":"94061961cbe6a700f4d3134859a0b6cc04764b5ee214c20e260a9225bf621748","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"94061961cbe6a700f4d3134859a0b6cc04764b5ee214c20e260a9225bf621748","first_computed_at":"2026-05-18T00:31:24.321753Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:31:24.321753Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+YAmVSG9s4RVzF8UAMIaYlS9CMBbF/w5/tMN11uAtF68dJSZWvxkFIKWXFZJDaryzc4zvVKAeYkJ0VfriLYkAg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:31:24.322252Z","signed_message":"canonical_sha256_bytes"},"source_id":"1711.00998","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5d3e0869931793c26fd2d649d5eaba698c9ba176a2e81a2cebb69b1a41ec0d57","sha256:e485bb0670b378835b389f3dd241a646edd92522bf925477c701f3c2baa252f8"],"state_sha256":"370a144d90eb127ee6a49ee0f15fb9a8408f03a12ff7406965e2188398a47a19"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JQNmCY94R1SGTT5pFMRTeCc42x3XXozph59yrYvmjxRHCQRH3c9xPybyCR9Csc8ky09JcTp+KaVgchtVW1q2Cg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T13:00:38.393056Z","bundle_sha256":"fcfd317601aab5e084667a5990c04504a03d937df82afb17eb3edfa459576d31"}}