{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:PWWHASSXZLPKPDZLIQIRWKWUS3","short_pith_number":"pith:PWWHASSX","canonical_record":{"source":{"id":"1107.3683","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-07-19T10:56:00Z","cross_cats_sorted":[],"title_canon_sha256":"ca8d3c3d9a20aa9a303563d640a6cef1f03cfe771b716486e18efffb304d3f42","abstract_canon_sha256":"c7ce6b6a4748bb1d5c8fa1b1f3ce911d4eedce989d26b5ff2392dea2f96bf935"},"schema_version":"1.0"},"canonical_sha256":"7dac704a57cadea78f2b44111b2ad496ff4b303253abc8442a9b2a66ba5ead79","source":{"kind":"arxiv","id":"1107.3683","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1107.3683","created_at":"2026-05-18T04:17:24Z"},{"alias_kind":"arxiv_version","alias_value":"1107.3683v1","created_at":"2026-05-18T04:17:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.3683","created_at":"2026-05-18T04:17:24Z"},{"alias_kind":"pith_short_12","alias_value":"PWWHASSXZLPK","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_16","alias_value":"PWWHASSXZLPKPDZL","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_8","alias_value":"PWWHASSX","created_at":"2026-05-18T12:26:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:PWWHASSXZLPKPDZLIQIRWKWUS3","target":"record","payload":{"canonical_record":{"source":{"id":"1107.3683","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-07-19T10:56:00Z","cross_cats_sorted":[],"title_canon_sha256":"ca8d3c3d9a20aa9a303563d640a6cef1f03cfe771b716486e18efffb304d3f42","abstract_canon_sha256":"c7ce6b6a4748bb1d5c8fa1b1f3ce911d4eedce989d26b5ff2392dea2f96bf935"},"schema_version":"1.0"},"canonical_sha256":"7dac704a57cadea78f2b44111b2ad496ff4b303253abc8442a9b2a66ba5ead79","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:17:24.190114Z","signature_b64":"K4B0eKaGq19v7wEzwD08wjkZ6kHp5LMBsdD0TlVxE0w8LCo+ZlPYh4TjM5QiATapvbbUORQ3vhcM+mdTxe1uDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7dac704a57cadea78f2b44111b2ad496ff4b303253abc8442a9b2a66ba5ead79","last_reissued_at":"2026-05-18T04:17:24.189369Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:17:24.189369Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1107.3683","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":"2026-05-18T04:17:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rcnGJXz0CFuU7evMGW3BJdWv9tf/HfFOlw2H3nN1PEV0Us9nTpaHenS54edub9naqcN1a4OxvpS0r07WvHwhBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T01:29:21.503291Z"},"content_sha256":"38de7ce081db51a71c740d61b13d8915fc7c8aafda25d7b607140039ecf6fbb4","schema_version":"1.0","event_id":"sha256:38de7ce081db51a71c740d61b13d8915fc7c8aafda25d7b607140039ecf6fbb4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:PWWHASSXZLPKPDZLIQIRWKWUS3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the approximation of a polytope by its dual $L_{p}$-centroid bodies","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Elisabeth M. Werner, Grigoris Paouris","submitted_at":"2011-07-19T10:56:00Z","abstract_excerpt":"We show that the rate of convergence on the approximation of volumes of a convex symmetric polytope P in R^n by its dual L_{p$-centroid bodies is independent of the geometry of P. In particular we show that if P has volume 1, lim_{p\\rightarrow \\infty} \\frac{p}{\\log{p}} (\\frac{|Z_{p}^{\\circ}(P)|}{|P^{\\circ}|} -1) = n^{2}.\n  We provide an application to the approximation of polytopes by uniformly convex sets."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.3683","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":"2026-05-18T04:17:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8gWpjo+/rgwWM9oazNkTKWiVMPdZUFhRZitNBLd+eTZ5O6XsI0uhvWQw0KxLC6FeWM83O8WGNpRjuXL+02rBCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T01:29:21.503940Z"},"content_sha256":"f89544fd634dcae2c3a712323009d1c4f45f66281d010d8b5609d8f3d7d75b3f","schema_version":"1.0","event_id":"sha256:f89544fd634dcae2c3a712323009d1c4f45f66281d010d8b5609d8f3d7d75b3f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PWWHASSXZLPKPDZLIQIRWKWUS3/bundle.json","state_url":"https://pith.science/pith/PWWHASSXZLPKPDZLIQIRWKWUS3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PWWHASSXZLPKPDZLIQIRWKWUS3/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":"2026-05-31T01:29:21Z","links":{"resolver":"https://pith.science/pith/PWWHASSXZLPKPDZLIQIRWKWUS3","bundle":"https://pith.science/pith/PWWHASSXZLPKPDZLIQIRWKWUS3/bundle.json","state":"https://pith.science/pith/PWWHASSXZLPKPDZLIQIRWKWUS3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PWWHASSXZLPKPDZLIQIRWKWUS3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:PWWHASSXZLPKPDZLIQIRWKWUS3","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":"c7ce6b6a4748bb1d5c8fa1b1f3ce911d4eedce989d26b5ff2392dea2f96bf935","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-07-19T10:56:00Z","title_canon_sha256":"ca8d3c3d9a20aa9a303563d640a6cef1f03cfe771b716486e18efffb304d3f42"},"schema_version":"1.0","source":{"id":"1107.3683","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1107.3683","created_at":"2026-05-18T04:17:24Z"},{"alias_kind":"arxiv_version","alias_value":"1107.3683v1","created_at":"2026-05-18T04:17:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.3683","created_at":"2026-05-18T04:17:24Z"},{"alias_kind":"pith_short_12","alias_value":"PWWHASSXZLPK","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_16","alias_value":"PWWHASSXZLPKPDZL","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_8","alias_value":"PWWHASSX","created_at":"2026-05-18T12:26:39Z"}],"graph_snapshots":[{"event_id":"sha256:f89544fd634dcae2c3a712323009d1c4f45f66281d010d8b5609d8f3d7d75b3f","target":"graph","created_at":"2026-05-18T04:17:24Z","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 that the rate of convergence on the approximation of volumes of a convex symmetric polytope P in R^n by its dual L_{p$-centroid bodies is independent of the geometry of P. In particular we show that if P has volume 1, lim_{p\\rightarrow \\infty} \\frac{p}{\\log{p}} (\\frac{|Z_{p}^{\\circ}(P)|}{|P^{\\circ}|} -1) = n^{2}.\n  We provide an application to the approximation of polytopes by uniformly convex sets.","authors_text":"Elisabeth M. Werner, Grigoris Paouris","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-07-19T10:56:00Z","title":"On the approximation of a polytope by its dual $L_{p}$-centroid bodies"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.3683","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":"sha256:38de7ce081db51a71c740d61b13d8915fc7c8aafda25d7b607140039ecf6fbb4","target":"record","created_at":"2026-05-18T04:17:24Z","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":"c7ce6b6a4748bb1d5c8fa1b1f3ce911d4eedce989d26b5ff2392dea2f96bf935","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-07-19T10:56:00Z","title_canon_sha256":"ca8d3c3d9a20aa9a303563d640a6cef1f03cfe771b716486e18efffb304d3f42"},"schema_version":"1.0","source":{"id":"1107.3683","kind":"arxiv","version":1}},"canonical_sha256":"7dac704a57cadea78f2b44111b2ad496ff4b303253abc8442a9b2a66ba5ead79","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7dac704a57cadea78f2b44111b2ad496ff4b303253abc8442a9b2a66ba5ead79","first_computed_at":"2026-05-18T04:17:24.189369Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:17:24.189369Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"K4B0eKaGq19v7wEzwD08wjkZ6kHp5LMBsdD0TlVxE0w8LCo+ZlPYh4TjM5QiATapvbbUORQ3vhcM+mdTxe1uDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:17:24.190114Z","signed_message":"canonical_sha256_bytes"},"source_id":"1107.3683","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:38de7ce081db51a71c740d61b13d8915fc7c8aafda25d7b607140039ecf6fbb4","sha256:f89544fd634dcae2c3a712323009d1c4f45f66281d010d8b5609d8f3d7d75b3f"],"state_sha256":"8552b95604c52a1647efc2066acc6d131932e95a4b6f845c4ffb59cf77264535"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7a2oFQ0LAIe1G8XkVN48YDLDjCkzzSzwCvDawIza/D4YOJmyGWznVC63cOINCPob286imHbZ6hk/JrlHdk39Aw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T01:29:21.507164Z","bundle_sha256":"2dd65a93188972f36599e69f93ee2592b7241193ce23c39f36a7d55415de7bdc"}}