{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:LNIMZS2C7O4L5UQ6WEVJW3FPAD","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":"e3f86efe59a784c44ae739897b4b35621da697e98f266edaa7522971c51463ad","cross_cats_sorted":["math.FA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.MG","submitted_at":"2023-01-15T16:16:01Z","title_canon_sha256":"16e3fcaee91f499fbc31aa56d5efa015b15ce25a40690f3c170a76beff199146"},"schema_version":"1.0","source":{"id":"2301.06131","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2301.06131","created_at":"2026-07-05T05:33:28Z"},{"alias_kind":"arxiv_version","alias_value":"2301.06131v1","created_at":"2026-07-05T05:33:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2301.06131","created_at":"2026-07-05T05:33:28Z"},{"alias_kind":"pith_short_12","alias_value":"LNIMZS2C7O4L","created_at":"2026-07-05T05:33:28Z"},{"alias_kind":"pith_short_16","alias_value":"LNIMZS2C7O4L5UQ6","created_at":"2026-07-05T05:33:28Z"},{"alias_kind":"pith_short_8","alias_value":"LNIMZS2C","created_at":"2026-07-05T05:33:28Z"}],"graph_snapshots":[{"event_id":"sha256:00162780cea09b3c4968be6368ecb0a6c4d2c70e1657628e722ebdee078a37da","target":"graph","created_at":"2026-07-05T05:33:28Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2301.06131/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Our purpose here is to give an overview of known results and open questions concerning the volume product ${\\mathcal P}(K)=\\min_{z\\in K}{\\rm vol}(K){\\rm vol}((K-z)^*)$ of a convex body $K$ in ${\\mathbb R}^n$. We present a number of upper and lower bounds for ${\\mathcal P}(K)$, in particular, we discuss the Mahler's conjecture on the lower bound of ${\\mathcal P}(K)$, which is still open. We also show connections of ${\\mathcal P}(K)$ with different parts of modern mathematics, including Geometric Number Theory, Convex Geometry, Analysis, Harmonic Analysis as well as Systolic and Symplectic Geome","authors_text":"Artem Zvavitch, Mathieu Meyer, Matthieu Fradelizi","cross_cats":["math.FA"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.MG","submitted_at":"2023-01-15T16:16:01Z","title":"Volume Product"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2301.06131","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":"sha256:61b71135667a6504303e1676f3a2c7e7095130784d09aa6cd97952c4189a5e5e","target":"record","created_at":"2026-07-05T05:33:28Z","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":"e3f86efe59a784c44ae739897b4b35621da697e98f266edaa7522971c51463ad","cross_cats_sorted":["math.FA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.MG","submitted_at":"2023-01-15T16:16:01Z","title_canon_sha256":"16e3fcaee91f499fbc31aa56d5efa015b15ce25a40690f3c170a76beff199146"},"schema_version":"1.0","source":{"id":"2301.06131","kind":"arxiv","version":1}},"canonical_sha256":"5b50cccb42fbb8bed21eb12a9b6caf00d7b9171d75eaab44f5d57e6c1ce6acf0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5b50cccb42fbb8bed21eb12a9b6caf00d7b9171d75eaab44f5d57e6c1ce6acf0","first_computed_at":"2026-07-05T05:33:28.222076Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T05:33:28.222076Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"MoK/Q4fA5eBqFg8iaz3DcmDoPuWp/H9LjYqqRv6rlsE53GltbfpOEP3ASA+2CMI0SegF6Dp2WuMAJWZAgMiCDQ==","signature_status":"signed_v1","signed_at":"2026-07-05T05:33:28.222530Z","signed_message":"canonical_sha256_bytes"},"source_id":"2301.06131","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:61b71135667a6504303e1676f3a2c7e7095130784d09aa6cd97952c4189a5e5e","sha256:00162780cea09b3c4968be6368ecb0a6c4d2c70e1657628e722ebdee078a37da"],"state_sha256":"cb9f1daf7b98fb8206c066e4687ba4e4ff70b1f1744bf4b3b5fb9323bf961f9b"}