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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 $$ n \\leq \\frac{c}{dt} \\ vol_{d}(K^{t} \\setminus K) $$"},"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":"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"},"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"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"math/9609206","created_at":"2026-05-18T01:38:23.022336+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/9609206v1","created_at":"2026-05-18T01:38:23.022336+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9609206","created_at":"2026-05-18T01:38:23.022336+00:00"},{"alias_kind":"pith_short_12","alias_value":"3CQAVPR3WF3B","created_at":"2026-05-18T12:25:47.700082+00:00"},{"alias_kind":"pith_short_16","alias_value":"3CQAVPR3WF3BHCHD","created_at":"2026-05-18T12:25:47.700082+00:00"},{"alias_kind":"pith_short_8","alias_value":"3CQAVPR3","created_at":"2026-05-18T12:25:47.700082+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/3CQAVPR3WF3BHCHD2G3MQLUJKX","json":"https://pith.science/pith/3CQAVPR3WF3BHCHD2G3MQLUJKX.json","graph_json":"https://pith.science/api/pith-number/3CQAVPR3WF3BHCHD2G3MQLUJKX/graph.json","events_json":"https://pith.science/api/pith-number/3CQAVPR3WF3BHCHD2G3MQLUJKX/events.json","paper":"https://pith.science/paper/3CQAVPR3"},"agent_actions":{"view_html":"https://pith.science/pith/3CQAVPR3WF3BHCHD2G3MQLUJKX","download_json":"https://pith.science/pith/3CQAVPR3WF3BHCHD2G3MQLUJKX.json","view_paper":"https://pith.science/paper/3CQAVPR3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/9609206&json=true","fetch_graph":"https://pith.science/api/pith-number/3CQAVPR3WF3BHCHD2G3MQLUJKX/graph.json","fetch_events":"https://pith.science/api/pith-number/3CQAVPR3WF3BHCHD2G3MQLUJKX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3CQAVPR3WF3BHCHD2G3MQLUJKX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3CQAVPR3WF3BHCHD2G3MQLUJKX/action/storage_attestation","attest_author":"https://pith.science/pith/3CQAVPR3WF3BHCHD2G3MQLUJKX/action/author_attestation","sign_citation":"https://pith.science/pith/3CQAVPR3WF3BHCHD2G3MQLUJKX/action/citation_signature","submit_replication":"https://pith.science/pith/3CQAVPR3WF3BHCHD2G3MQLUJKX/action/replication_record"}},"created_at":"2026-05-18T01:38:23.022336+00:00","updated_at":"2026-05-18T01:38:23.022336+00:00"}