{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:KNKP5M4ZQSMLOMR24HGPO7XQ7X","short_pith_number":"pith:KNKP5M4Z","canonical_record":{"source":{"id":"0905.3514","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2009-05-21T15:59:47Z","cross_cats_sorted":[],"title_canon_sha256":"fcf0e61764a694166e8445cbf0d648e43762689b414c33c8ba5937b0e88fc8d2","abstract_canon_sha256":"bbc7758665dff3ff9bce0e9c3f848603e036d25304cdd84042e46c7ff7db7900"},"schema_version":"1.0"},"canonical_sha256":"5354feb3998498b7323ae1ccf77ef0fdee85b6bb2993e72fc2de8e24b4c0a323","source":{"kind":"arxiv","id":"0905.3514","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0905.3514","created_at":"2026-05-18T04:38:58Z"},{"alias_kind":"arxiv_version","alias_value":"0905.3514v2","created_at":"2026-05-18T04:38:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0905.3514","created_at":"2026-05-18T04:38:58Z"},{"alias_kind":"pith_short_12","alias_value":"KNKP5M4ZQSML","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_16","alias_value":"KNKP5M4ZQSMLOMR2","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_8","alias_value":"KNKP5M4Z","created_at":"2026-05-18T12:26:00Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:KNKP5M4ZQSMLOMR24HGPO7XQ7X","target":"record","payload":{"canonical_record":{"source":{"id":"0905.3514","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2009-05-21T15:59:47Z","cross_cats_sorted":[],"title_canon_sha256":"fcf0e61764a694166e8445cbf0d648e43762689b414c33c8ba5937b0e88fc8d2","abstract_canon_sha256":"bbc7758665dff3ff9bce0e9c3f848603e036d25304cdd84042e46c7ff7db7900"},"schema_version":"1.0"},"canonical_sha256":"5354feb3998498b7323ae1ccf77ef0fdee85b6bb2993e72fc2de8e24b4c0a323","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:38:58.337197Z","signature_b64":"TpVhM/gOLcPDGt0hZgBC4VioKWDg6oIXlSOxMwCE+YI0oXCpiVz37rj8RsTotJrTegM2zbCfgmmB4USJOUfhCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5354feb3998498b7323ae1ccf77ef0fdee85b6bb2993e72fc2de8e24b4c0a323","last_reissued_at":"2026-05-18T04:38:58.336656Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:38:58.336656Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0905.3514","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:38:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dZq1UPu/LTFZOTwe3n6BwFANJ6s4b6w9AxR9qFLDGyF6bfYvPINyM0gbk1Iy9lb5aMIVtxNNuNseAkhpajLwAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T12:23:28.818904Z"},"content_sha256":"76e2508d6764ceb23d45392adeda04f7e005f1a7c5bc4503fac37ea58d69f1c6","schema_version":"1.0","event_id":"sha256:76e2508d6764ceb23d45392adeda04f7e005f1a7c5bc4503fac37ea58d69f1c6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:KNKP5M4ZQSMLOMR24HGPO7XQ7X","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Containment and inscribed simplices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Daniel A. Klain","submitted_at":"2009-05-21T15:59:47Z","abstract_excerpt":"Let K and L be compact convex sets in R^n. The following two statements are shown to be equivalent: (i) For every polytope Q inside K having at most n+1 vertices, L contains a translate of Q. (ii) L contains a translate of K.\n  Let 1 <= d <= n-1. It is also shown that the following two statements are equivalent: (i) For every polytope Q inside K having at most d+1 vertices, L contains a translate of Q. (ii) For every d-dimensional subspace W, the orthogonal projection of the set L onto W contains a translate of the corresponding projection of the set K onto W.\n  It is then shown that, if K is "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0905.3514","kind":"arxiv","version":2},"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":"2026-05-18T04:38:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"N2RmD8ZbJ0kK05Fp2hJVfNYG1msEDAoxAXEXkM5Al5dGRcpf3okGPy5RPeVPOXvoPkyDSJAY3J3uaFgGPEW5DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T12:23:28.819248Z"},"content_sha256":"67cf637f5cc2530ae285be0e30c0d297495cbc933a74562fb3d81d3ac691c708","schema_version":"1.0","event_id":"sha256:67cf637f5cc2530ae285be0e30c0d297495cbc933a74562fb3d81d3ac691c708"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KNKP5M4ZQSMLOMR24HGPO7XQ7X/bundle.json","state_url":"https://pith.science/pith/KNKP5M4ZQSMLOMR24HGPO7XQ7X/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KNKP5M4ZQSMLOMR24HGPO7XQ7X/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":"2026-06-24T12:23:28Z","links":{"resolver":"https://pith.science/pith/KNKP5M4ZQSMLOMR24HGPO7XQ7X","bundle":"https://pith.science/pith/KNKP5M4ZQSMLOMR24HGPO7XQ7X/bundle.json","state":"https://pith.science/pith/KNKP5M4ZQSMLOMR24HGPO7XQ7X/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KNKP5M4ZQSMLOMR24HGPO7XQ7X/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:KNKP5M4ZQSMLOMR24HGPO7XQ7X","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":"bbc7758665dff3ff9bce0e9c3f848603e036d25304cdd84042e46c7ff7db7900","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2009-05-21T15:59:47Z","title_canon_sha256":"fcf0e61764a694166e8445cbf0d648e43762689b414c33c8ba5937b0e88fc8d2"},"schema_version":"1.0","source":{"id":"0905.3514","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0905.3514","created_at":"2026-05-18T04:38:58Z"},{"alias_kind":"arxiv_version","alias_value":"0905.3514v2","created_at":"2026-05-18T04:38:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0905.3514","created_at":"2026-05-18T04:38:58Z"},{"alias_kind":"pith_short_12","alias_value":"KNKP5M4ZQSML","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_16","alias_value":"KNKP5M4ZQSMLOMR2","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_8","alias_value":"KNKP5M4Z","created_at":"2026-05-18T12:26:00Z"}],"graph_snapshots":[{"event_id":"sha256:67cf637f5cc2530ae285be0e30c0d297495cbc933a74562fb3d81d3ac691c708","target":"graph","created_at":"2026-05-18T04:38:58Z","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 and L be compact convex sets in R^n. The following two statements are shown to be equivalent: (i) For every polytope Q inside K having at most n+1 vertices, L contains a translate of Q. (ii) L contains a translate of K.\n  Let 1 <= d <= n-1. It is also shown that the following two statements are equivalent: (i) For every polytope Q inside K having at most d+1 vertices, L contains a translate of Q. (ii) For every d-dimensional subspace W, the orthogonal projection of the set L onto W contains a translate of the corresponding projection of the set K onto W.\n  It is then shown that, if K is ","authors_text":"Daniel A. Klain","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2009-05-21T15:59:47Z","title":"Containment and inscribed simplices"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0905.3514","kind":"arxiv","version":2},"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":"sha256:76e2508d6764ceb23d45392adeda04f7e005f1a7c5bc4503fac37ea58d69f1c6","target":"record","created_at":"2026-05-18T04:38:58Z","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":"bbc7758665dff3ff9bce0e9c3f848603e036d25304cdd84042e46c7ff7db7900","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2009-05-21T15:59:47Z","title_canon_sha256":"fcf0e61764a694166e8445cbf0d648e43762689b414c33c8ba5937b0e88fc8d2"},"schema_version":"1.0","source":{"id":"0905.3514","kind":"arxiv","version":2}},"canonical_sha256":"5354feb3998498b7323ae1ccf77ef0fdee85b6bb2993e72fc2de8e24b4c0a323","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5354feb3998498b7323ae1ccf77ef0fdee85b6bb2993e72fc2de8e24b4c0a323","first_computed_at":"2026-05-18T04:38:58.336656Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:38:58.336656Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"TpVhM/gOLcPDGt0hZgBC4VioKWDg6oIXlSOxMwCE+YI0oXCpiVz37rj8RsTotJrTegM2zbCfgmmB4USJOUfhCg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:38:58.337197Z","signed_message":"canonical_sha256_bytes"},"source_id":"0905.3514","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:76e2508d6764ceb23d45392adeda04f7e005f1a7c5bc4503fac37ea58d69f1c6","sha256:67cf637f5cc2530ae285be0e30c0d297495cbc933a74562fb3d81d3ac691c708"],"state_sha256":"4122ec7b8116149c2392d8685763f5c55218839cd106fb69902352b5187a80ee"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0tfs1qHpjogxpTEIGFiwRUoPOzNtDBlzP5Ncio+a1Yw+HEoLsGxXgSMO8vo7K+fBmAv86pXpVloo450sjTGzBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T12:23:28.821185Z","bundle_sha256":"68a4da2ab25d510017fc9f3cfc3f3bd93ea6871a06e9a8154aa6b232eda3341b"}}