{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:1996:3CQAVPR3WF3BHCHD2G3MQLUJKX","short_pith_number":"pith:3CQAVPR3","canonical_record":{"source":{"id":"math/9609206","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.MG","submitted_at":"1996-09-05T00:00:00Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"7a3f96428f993c298720aea51330bbd90111271ccfab8b15a061dda5528deac8","abstract_canon_sha256":"cabeac6a0390766d63e70cc0fe976c87c8ec5c7493c88d8cfbf0d9bd5c297b0f"},"schema_version":"1.0"},"canonical_sha256":"d8a00abe3bb1761388e3d1b6c82e8955eaddd3388bd0e4ff972a1d335711bbd9","source":{"kind":"arxiv","id":"math/9609206","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9609206","created_at":"2026-05-18T01:38:23Z"},{"alias_kind":"arxiv_version","alias_value":"math/9609206v1","created_at":"2026-05-18T01:38:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9609206","created_at":"2026-05-18T01:38:23Z"},{"alias_kind":"pith_short_12","alias_value":"3CQAVPR3WF3B","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_16","alias_value":"3CQAVPR3WF3BHCHD","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_8","alias_value":"3CQAVPR3","created_at":"2026-05-18T12:25:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:1996:3CQAVPR3WF3BHCHD2G3MQLUJKX","target":"record","payload":{"canonical_record":{"source":{"id":"math/9609206","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.MG","submitted_at":"1996-09-05T00:00:00Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"7a3f96428f993c298720aea51330bbd90111271ccfab8b15a061dda5528deac8","abstract_canon_sha256":"cabeac6a0390766d63e70cc0fe976c87c8ec5c7493c88d8cfbf0d9bd5c297b0f"},"schema_version":"1.0"},"canonical_sha256":"d8a00abe3bb1761388e3d1b6c82e8955eaddd3388bd0e4ff972a1d335711bbd9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:38:23.022950Z","signature_b64":"Faen4fwz0zuiwTXvRo2XkfLlSiO6fsqxBZnnwBDNViT/1vNHv1BdGUnyJ6gNnOG09T8V/1IBqjhCngYxpJt6Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d8a00abe3bb1761388e3d1b6c82e8955eaddd3388bd0e4ff972a1d335711bbd9","last_reissued_at":"2026-05-18T01:38:23.022201Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:38:23.022201Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/9609206","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":"2026-05-18T01:38:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xDJJLnsTbfq+jkJnHuQcFXRA1eACBqki+K9fENYF2RX6TtQGCWN5fzXHufAzKTfX4uYXvmqxFLud1BXdwQyZBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T09:23:23.690597Z"},"content_sha256":"5e5033fc98ba7d9f2a9c0d9ea449fa24154812c691346e11b573fdfb0c85d87c","schema_version":"1.0","event_id":"sha256:5e5033fc98ba7d9f2a9c0d9ea449fa24154812c691346e11b573fdfb0c85d87c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:1996:3CQAVPR3WF3BHCHD2G3MQLUJKX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Floating body, illumination body, and polytopal approximation","license":"","headline":"","cross_cats":["math.FA"],"primary_cat":"math.MG","authors_text":"Carsten Sch\\\"utt","submitted_at":"1996-09-05T00:00:00Z","abstract_excerpt":"Let $K$ be a convex body in $\\Bbb R^{d}$ and $K_{t}$ its floating bodies. There is a polytope with at most $n$ vertices that satisfies $$ K_{t} \\subset P_{n} \\subset K $$ where $$ n \\leq e^{16d} \\frac{vol_{d}(K \\setminus K_{t})}{t\\ vol_{d}(B_{2}^{d})} $$ Let $K^{t}$ be the illumination bodies of $K$ and $Q_{n}$ a polytope that contains $K$ and has at most $n$ $d-1$-dimensional faces. Then $$ vol_{d}(K^{t} \\setminus K) \\leq cd^{4} vol_{d}(Q_{n} \\setminus K) $$ where $$ n \\leq \\frac{c}{dt} \\ vol_{d}(K^{t} \\setminus K) $$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9609206","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":"2026-05-18T01:38:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BYHbl1kdRg/TE7PaQRMVfjr2LYeH3lcj1J/m7kES2su4vvl0250XjovObA/ARUZPhlIYleG2F0R1WGSqffFYBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T09:23:23.690947Z"},"content_sha256":"72638eb88d33db236ed59d5325d100a1eba2c8da978a73dc1a7aef68c4482461","schema_version":"1.0","event_id":"sha256:72638eb88d33db236ed59d5325d100a1eba2c8da978a73dc1a7aef68c4482461"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3CQAVPR3WF3BHCHD2G3MQLUJKX/bundle.json","state_url":"https://pith.science/pith/3CQAVPR3WF3BHCHD2G3MQLUJKX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3CQAVPR3WF3BHCHD2G3MQLUJKX/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":"2026-06-01T09:23:23Z","links":{"resolver":"https://pith.science/pith/3CQAVPR3WF3BHCHD2G3MQLUJKX","bundle":"https://pith.science/pith/3CQAVPR3WF3BHCHD2G3MQLUJKX/bundle.json","state":"https://pith.science/pith/3CQAVPR3WF3BHCHD2G3MQLUJKX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3CQAVPR3WF3BHCHD2G3MQLUJKX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:1996:3CQAVPR3WF3BHCHD2G3MQLUJKX","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":"cabeac6a0390766d63e70cc0fe976c87c8ec5c7493c88d8cfbf0d9bd5c297b0f","cross_cats_sorted":["math.FA"],"license":"","primary_cat":"math.MG","submitted_at":"1996-09-05T00:00:00Z","title_canon_sha256":"7a3f96428f993c298720aea51330bbd90111271ccfab8b15a061dda5528deac8"},"schema_version":"1.0","source":{"id":"math/9609206","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9609206","created_at":"2026-05-18T01:38:23Z"},{"alias_kind":"arxiv_version","alias_value":"math/9609206v1","created_at":"2026-05-18T01:38:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9609206","created_at":"2026-05-18T01:38:23Z"},{"alias_kind":"pith_short_12","alias_value":"3CQAVPR3WF3B","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_16","alias_value":"3CQAVPR3WF3BHCHD","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_8","alias_value":"3CQAVPR3","created_at":"2026-05-18T12:25:47Z"}],"graph_snapshots":[{"event_id":"sha256:72638eb88d33db236ed59d5325d100a1eba2c8da978a73dc1a7aef68c4482461","target":"graph","created_at":"2026-05-18T01:38:23Z","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":"Let $K$ be a convex body in $\\Bbb R^{d}$ and $K_{t}$ its floating bodies. There is a polytope with at most $n$ vertices that satisfies $$ K_{t} \\subset P_{n} \\subset K $$ where $$ n \\leq e^{16d} \\frac{vol_{d}(K \\setminus K_{t})}{t\\ vol_{d}(B_{2}^{d})} $$ Let $K^{t}$ be the illumination bodies of $K$ and $Q_{n}$ a polytope that contains $K$ and has at most $n$ $d-1$-dimensional faces. Then $$ vol_{d}(K^{t} \\setminus K) \\leq cd^{4} vol_{d}(Q_{n} \\setminus K) $$ where $$ n \\leq \\frac{c}{dt} \\ vol_{d}(K^{t} \\setminus K) $$","authors_text":"Carsten Sch\\\"utt","cross_cats":["math.FA"],"headline":"","license":"","primary_cat":"math.MG","submitted_at":"1996-09-05T00:00:00Z","title":"Floating body, illumination body, and polytopal approximation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9609206","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":"sha256:5e5033fc98ba7d9f2a9c0d9ea449fa24154812c691346e11b573fdfb0c85d87c","target":"record","created_at":"2026-05-18T01:38:23Z","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":"cabeac6a0390766d63e70cc0fe976c87c8ec5c7493c88d8cfbf0d9bd5c297b0f","cross_cats_sorted":["math.FA"],"license":"","primary_cat":"math.MG","submitted_at":"1996-09-05T00:00:00Z","title_canon_sha256":"7a3f96428f993c298720aea51330bbd90111271ccfab8b15a061dda5528deac8"},"schema_version":"1.0","source":{"id":"math/9609206","kind":"arxiv","version":1}},"canonical_sha256":"d8a00abe3bb1761388e3d1b6c82e8955eaddd3388bd0e4ff972a1d335711bbd9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d8a00abe3bb1761388e3d1b6c82e8955eaddd3388bd0e4ff972a1d335711bbd9","first_computed_at":"2026-05-18T01:38:23.022201Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:38:23.022201Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Faen4fwz0zuiwTXvRo2XkfLlSiO6fsqxBZnnwBDNViT/1vNHv1BdGUnyJ6gNnOG09T8V/1IBqjhCngYxpJt6Cg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:38:23.022950Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/9609206","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5e5033fc98ba7d9f2a9c0d9ea449fa24154812c691346e11b573fdfb0c85d87c","sha256:72638eb88d33db236ed59d5325d100a1eba2c8da978a73dc1a7aef68c4482461"],"state_sha256":"bb8f7b0b7b58bfa36fe6df5563ee1173af82dba7d70212c2d15815775abdf917"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"euCMCow8v1aZeXG3OtnwkHpZ8/gP+qCguaY2eloTgsvVH0S5RgxGrS5kq8XbPDf0UlJWKcv+d41QIhxAYFHcDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T09:23:23.692952Z","bundle_sha256":"38b03941c2689314d3c06d73ae0f96e99f7f4c4150c92f39c19a7f35cc555c77"}}