{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:BQZVGS6DJV5WWNSQT7Z4QIMPYF","short_pith_number":"pith:BQZVGS6D","schema_version":"1.0","canonical_sha256":"0c33534bc34d7b6b36509ff3c8218fc17a81cade9d0c8ca47ee9cc71bb5286ef","source":{"kind":"arxiv","id":"1409.4347","version":2},"attestation_state":"computed","paper":{"title":"On the shape of a convex body with respect to its second projection body","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Christos Saroglou","submitted_at":"2014-09-15T17:43:25Z","abstract_excerpt":"We prove results relative to the problem of finding sharp bounds for the affine invariant $P(K)=V(\\Pi K)/V^{d-1}(K)$. Namely, we prove that if $K$ is a 3-dimensional zonoid of volume 1, then its second projection body $\\Pi^2K$ is contained in 8K, while if $K$ is any symmetric 3-dimensional convex body of volume 1, then $\\Pi^2K$ contains 6K. Both inclusions are sharp. Consequences of these results include a stronger version of a reverse isoperimetric inequality for 3-dimensional zonoids-established by the author in a previous work, a reduction for the 3-dimensional Petty conjecture to another i"},"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":"1409.4347","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2014-09-15T17:43:25Z","cross_cats_sorted":[],"title_canon_sha256":"11bf6b73ce47e512084ea16e996ece5762db50d2dcdaab08414ecebed543407e","abstract_canon_sha256":"1a4515d505e56a58610b47951040a5dd6fcd30f81d33f04b4dcb8db39b455d09"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:42:45.621162Z","signature_b64":"7iwjoFqy5D2uOrzkYylkEB4n9Qo/mx0oDZqt2xnXUG2VZHPtg0tUJeRb6M5sQH/fEou825Q2x3bFVmfqNyNvCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0c33534bc34d7b6b36509ff3c8218fc17a81cade9d0c8ca47ee9cc71bb5286ef","last_reissued_at":"2026-05-18T02:42:45.620688Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:42:45.620688Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the shape of a convex body with respect to its second projection body","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Christos Saroglou","submitted_at":"2014-09-15T17:43:25Z","abstract_excerpt":"We prove results relative to the problem of finding sharp bounds for the affine invariant $P(K)=V(\\Pi K)/V^{d-1}(K)$. Namely, we prove that if $K$ is a 3-dimensional zonoid of volume 1, then its second projection body $\\Pi^2K$ is contained in 8K, while if $K$ is any symmetric 3-dimensional convex body of volume 1, then $\\Pi^2K$ contains 6K. Both inclusions are sharp. Consequences of these results include a stronger version of a reverse isoperimetric inequality for 3-dimensional zonoids-established by the author in a previous work, a reduction for the 3-dimensional Petty conjecture to another i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.4347","kind":"arxiv","version":2},"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":"1409.4347","created_at":"2026-05-18T02:42:45.620758+00:00"},{"alias_kind":"arxiv_version","alias_value":"1409.4347v2","created_at":"2026-05-18T02:42:45.620758+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.4347","created_at":"2026-05-18T02:42:45.620758+00:00"},{"alias_kind":"pith_short_12","alias_value":"BQZVGS6DJV5W","created_at":"2026-05-18T12:28:22.404517+00:00"},{"alias_kind":"pith_short_16","alias_value":"BQZVGS6DJV5WWNSQ","created_at":"2026-05-18T12:28:22.404517+00:00"},{"alias_kind":"pith_short_8","alias_value":"BQZVGS6D","created_at":"2026-05-18T12:28:22.404517+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/BQZVGS6DJV5WWNSQT7Z4QIMPYF","json":"https://pith.science/pith/BQZVGS6DJV5WWNSQT7Z4QIMPYF.json","graph_json":"https://pith.science/api/pith-number/BQZVGS6DJV5WWNSQT7Z4QIMPYF/graph.json","events_json":"https://pith.science/api/pith-number/BQZVGS6DJV5WWNSQT7Z4QIMPYF/events.json","paper":"https://pith.science/paper/BQZVGS6D"},"agent_actions":{"view_html":"https://pith.science/pith/BQZVGS6DJV5WWNSQT7Z4QIMPYF","download_json":"https://pith.science/pith/BQZVGS6DJV5WWNSQT7Z4QIMPYF.json","view_paper":"https://pith.science/paper/BQZVGS6D","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1409.4347&json=true","fetch_graph":"https://pith.science/api/pith-number/BQZVGS6DJV5WWNSQT7Z4QIMPYF/graph.json","fetch_events":"https://pith.science/api/pith-number/BQZVGS6DJV5WWNSQT7Z4QIMPYF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BQZVGS6DJV5WWNSQT7Z4QIMPYF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BQZVGS6DJV5WWNSQT7Z4QIMPYF/action/storage_attestation","attest_author":"https://pith.science/pith/BQZVGS6DJV5WWNSQT7Z4QIMPYF/action/author_attestation","sign_citation":"https://pith.science/pith/BQZVGS6DJV5WWNSQT7Z4QIMPYF/action/citation_signature","submit_replication":"https://pith.science/pith/BQZVGS6DJV5WWNSQT7Z4QIMPYF/action/replication_record"}},"created_at":"2026-05-18T02:42:45.620758+00:00","updated_at":"2026-05-18T02:42:45.620758+00:00"}