{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:3Y7YUVRXVYGDPV3RSSKXGPEEO4","short_pith_number":"pith:3Y7YUVRX","canonical_record":{"source":{"id":"1207.1342","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-07-05T19:50:17Z","cross_cats_sorted":["math.DG","math.SG"],"title_canon_sha256":"8aad08e163772206814294355e40cb1f756d317b7754121c006f887d56cfaf01","abstract_canon_sha256":"6db71236438c83c70ab40b9adf6ed051ceb94b291557d7095d7132b9a623d162"},"schema_version":"1.0"},"canonical_sha256":"de3f8a5637ae0c37d7719495733c8477172945b6f9ae2845098420cd4b8ae755","source":{"kind":"arxiv","id":"1207.1342","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.1342","created_at":"2026-05-18T00:49:52Z"},{"alias_kind":"arxiv_version","alias_value":"1207.1342v3","created_at":"2026-05-18T00:49:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.1342","created_at":"2026-05-18T00:49:52Z"},{"alias_kind":"pith_short_12","alias_value":"3Y7YUVRXVYGD","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_16","alias_value":"3Y7YUVRXVYGDPV3R","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_8","alias_value":"3Y7YUVRX","created_at":"2026-05-18T12:26:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:3Y7YUVRXVYGDPV3RSSKXGPEEO4","target":"record","payload":{"canonical_record":{"source":{"id":"1207.1342","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-07-05T19:50:17Z","cross_cats_sorted":["math.DG","math.SG"],"title_canon_sha256":"8aad08e163772206814294355e40cb1f756d317b7754121c006f887d56cfaf01","abstract_canon_sha256":"6db71236438c83c70ab40b9adf6ed051ceb94b291557d7095d7132b9a623d162"},"schema_version":"1.0"},"canonical_sha256":"de3f8a5637ae0c37d7719495733c8477172945b6f9ae2845098420cd4b8ae755","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:49:52.957344Z","signature_b64":"DyfIZVuZ4jEL+5UrjdVc2TdLENcUUebpYeXL2ti4ur7lb/BoSJ10dMcCqlc/firYtpayE7xeayF8Y/WApX3tCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"de3f8a5637ae0c37d7719495733c8477172945b6f9ae2845098420cd4b8ae755","last_reissued_at":"2026-05-18T00:49:52.956822Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:49:52.956822Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1207.1342","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:49:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"x6c0+FspH8zTagfmhRqbPj0pMBdWxjbNSga/BavTrZ6NR+hU8zmeHjJeVSrSecBSEOWRvbux7PByflK/mi3hCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T10:59:11.013981Z"},"content_sha256":"b1b7d4eb97b2bd0673f2f95a566dc4f6a196e89789042f7941bc9b212d40958d","schema_version":"1.0","event_id":"sha256:b1b7d4eb97b2bd0673f2f95a566dc4f6a196e89789042f7941bc9b212d40958d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:3Y7YUVRXVYGDPV3RSSKXGPEEO4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Approximability of convex bodies and volume entropy in Hilbert geometry","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG","math.SG"],"primary_cat":"math.MG","authors_text":"Constantin Vernicos","submitted_at":"2012-07-05T19:50:17Z","abstract_excerpt":"The approximability of a convex body is a number which measures the difficulty to approximate that body by polytopes. We prove that twice the approximability is equal to the volume entropy for a Hilbert geometry in dimension two end three and that in higher dimension it is a lower bound of the entropy.\n  As a corollary we solve the entropy upper bound conjecture in dimension three and give a new proof in dimension two from the one found in Berck-Bernig-Vernicos (arXiv:0810.1123v2, published)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.1342","kind":"arxiv","version":3},"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":"2026-05-18T00:49:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0dv0cc566YWtLw0nyOd8j4vX+/WmqvrNZJonkBPFuzV5G5/VlCTaZHYIp9P+cjKOCulo4cfvi/MpI6+rQ6Z1DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T10:59:11.014347Z"},"content_sha256":"eb69f5fe940aece20a3a09a996c9f922614100ca2e626f431e5eeeae32e5bd76","schema_version":"1.0","event_id":"sha256:eb69f5fe940aece20a3a09a996c9f922614100ca2e626f431e5eeeae32e5bd76"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3Y7YUVRXVYGDPV3RSSKXGPEEO4/bundle.json","state_url":"https://pith.science/pith/3Y7YUVRXVYGDPV3RSSKXGPEEO4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3Y7YUVRXVYGDPV3RSSKXGPEEO4/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":"2026-06-26T10:59:11Z","links":{"resolver":"https://pith.science/pith/3Y7YUVRXVYGDPV3RSSKXGPEEO4","bundle":"https://pith.science/pith/3Y7YUVRXVYGDPV3RSSKXGPEEO4/bundle.json","state":"https://pith.science/pith/3Y7YUVRXVYGDPV3RSSKXGPEEO4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3Y7YUVRXVYGDPV3RSSKXGPEEO4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:3Y7YUVRXVYGDPV3RSSKXGPEEO4","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":"6db71236438c83c70ab40b9adf6ed051ceb94b291557d7095d7132b9a623d162","cross_cats_sorted":["math.DG","math.SG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-07-05T19:50:17Z","title_canon_sha256":"8aad08e163772206814294355e40cb1f756d317b7754121c006f887d56cfaf01"},"schema_version":"1.0","source":{"id":"1207.1342","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.1342","created_at":"2026-05-18T00:49:52Z"},{"alias_kind":"arxiv_version","alias_value":"1207.1342v3","created_at":"2026-05-18T00:49:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.1342","created_at":"2026-05-18T00:49:52Z"},{"alias_kind":"pith_short_12","alias_value":"3Y7YUVRXVYGD","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_16","alias_value":"3Y7YUVRXVYGDPV3R","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_8","alias_value":"3Y7YUVRX","created_at":"2026-05-18T12:26:53Z"}],"graph_snapshots":[{"event_id":"sha256:eb69f5fe940aece20a3a09a996c9f922614100ca2e626f431e5eeeae32e5bd76","target":"graph","created_at":"2026-05-18T00:49:52Z","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":"The approximability of a convex body is a number which measures the difficulty to approximate that body by polytopes. We prove that twice the approximability is equal to the volume entropy for a Hilbert geometry in dimension two end three and that in higher dimension it is a lower bound of the entropy.\n  As a corollary we solve the entropy upper bound conjecture in dimension three and give a new proof in dimension two from the one found in Berck-Bernig-Vernicos (arXiv:0810.1123v2, published).","authors_text":"Constantin Vernicos","cross_cats":["math.DG","math.SG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-07-05T19:50:17Z","title":"Approximability of convex bodies and volume entropy in Hilbert geometry"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.1342","kind":"arxiv","version":3},"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:b1b7d4eb97b2bd0673f2f95a566dc4f6a196e89789042f7941bc9b212d40958d","target":"record","created_at":"2026-05-18T00:49:52Z","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":"6db71236438c83c70ab40b9adf6ed051ceb94b291557d7095d7132b9a623d162","cross_cats_sorted":["math.DG","math.SG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-07-05T19:50:17Z","title_canon_sha256":"8aad08e163772206814294355e40cb1f756d317b7754121c006f887d56cfaf01"},"schema_version":"1.0","source":{"id":"1207.1342","kind":"arxiv","version":3}},"canonical_sha256":"de3f8a5637ae0c37d7719495733c8477172945b6f9ae2845098420cd4b8ae755","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"de3f8a5637ae0c37d7719495733c8477172945b6f9ae2845098420cd4b8ae755","first_computed_at":"2026-05-18T00:49:52.956822Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:49:52.956822Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"DyfIZVuZ4jEL+5UrjdVc2TdLENcUUebpYeXL2ti4ur7lb/BoSJ10dMcCqlc/firYtpayE7xeayF8Y/WApX3tCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:49:52.957344Z","signed_message":"canonical_sha256_bytes"},"source_id":"1207.1342","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b1b7d4eb97b2bd0673f2f95a566dc4f6a196e89789042f7941bc9b212d40958d","sha256:eb69f5fe940aece20a3a09a996c9f922614100ca2e626f431e5eeeae32e5bd76"],"state_sha256":"e34685e8d14ee9f9fe40265b88e0f25f9284703a3f5da2a6d9642cd7d53f6c1e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cpOzDVpW0l8AvOH+1BMBnKN4mUECGfw2ZAFw4nQ6Q7AWG7CrlWbWEtvBoA2r1VcVweZ87w6DMpGwCv73wH8zCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T10:59:11.016308Z","bundle_sha256":"55d5416bbc78a062912e20a056d13a534151b5d041bbea507fded2ac2ecacdc8"}}