{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:5ZJONB4OJL3FOVVPYVWWHUD4VY","short_pith_number":"pith:5ZJONB4O","canonical_record":{"source":{"id":"0903.4842","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2009-03-27T16:26:46Z","cross_cats_sorted":[],"title_canon_sha256":"1149651d7c13a2783e26aeb19301e12d7bbffdc8346cf13ca1cebfd81b66731f","abstract_canon_sha256":"33aba11b2b12b4666bcdf28af933f5bd7cabd76b8bba4864715ee5165fdc059c"},"schema_version":"1.0"},"canonical_sha256":"ee52e6878e4af65756afc56d63d07cae0a7b3b095f5275997489f078c7e7ca65","source":{"kind":"arxiv","id":"0903.4842","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0903.4842","created_at":"2026-05-18T04:12:10Z"},{"alias_kind":"arxiv_version","alias_value":"0903.4842v1","created_at":"2026-05-18T04:12:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0903.4842","created_at":"2026-05-18T04:12:10Z"},{"alias_kind":"pith_short_12","alias_value":"5ZJONB4OJL3F","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"5ZJONB4OJL3FOVVP","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"5ZJONB4O","created_at":"2026-05-18T12:25:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:5ZJONB4OJL3FOVVPYVWWHUD4VY","target":"record","payload":{"canonical_record":{"source":{"id":"0903.4842","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2009-03-27T16:26:46Z","cross_cats_sorted":[],"title_canon_sha256":"1149651d7c13a2783e26aeb19301e12d7bbffdc8346cf13ca1cebfd81b66731f","abstract_canon_sha256":"33aba11b2b12b4666bcdf28af933f5bd7cabd76b8bba4864715ee5165fdc059c"},"schema_version":"1.0"},"canonical_sha256":"ee52e6878e4af65756afc56d63d07cae0a7b3b095f5275997489f078c7e7ca65","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:12:10.731556Z","signature_b64":"z2ne3S3iHpfYVcpC1Y6xbC822s+Y0vd0J/3VKT5VM8IjX1ptcWvvgkvaTDes665M89qeaaCl3r6Q9FrhcZcgAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ee52e6878e4af65756afc56d63d07cae0a7b3b095f5275997489f078c7e7ca65","last_reissued_at":"2026-05-18T04:12:10.730995Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:12:10.730995Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0903.4842","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":"2026-05-18T04:12:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LpCMn/dJ+EYv6/35IbP+HjsjOucPu8L9S60mLeAx1Qxt+yPb5N54EYdaGRHi6L3DEmOCp7CtZn/TwlRMaFnzBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T03:00:21.748606Z"},"content_sha256":"cf7ea8d6e84fff6062edd2206f4095b8a455451617929561bb68b628123a5de5","schema_version":"1.0","event_id":"sha256:cf7ea8d6e84fff6062edd2206f4095b8a455451617929561bb68b628123a5de5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:5ZJONB4OJL3FOVVPYVWWHUD4VY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Covering convex bodies by cylinders and lattice points by flats","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Alexander Litvak, Karoly Bezdek","submitted_at":"2009-03-27T16:26:46Z","abstract_excerpt":"In connection with an unsolved problem of Bang (1951) we give a lower bound for the sum of the base volumes of cylinders covering a d-dimensional convex body in terms of the relevant basic measures of the given convex body. As an application we establish lower bounds on the number of k-dimensional flats (i.e. translates of k-dimensional linear subspaces) needed to cover all the integer points of a given convex body in d-dimensional Euclidean space for 0<k<d."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0903.4842","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":"2026-05-18T04:12:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"prrqNdedjCdPwKDeo+WFFUWsPfUJH3ktVPGXIDad9k20CW8eTZbzJeTEApqFL4XmJu6IWvnEa1kk6SB9PR78DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T03:00:21.748935Z"},"content_sha256":"c5f0bc5c42b39fd20ac32bb8f2ecd1d2e53378ee2cd1208da004ea9e1be5cd45","schema_version":"1.0","event_id":"sha256:c5f0bc5c42b39fd20ac32bb8f2ecd1d2e53378ee2cd1208da004ea9e1be5cd45"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5ZJONB4OJL3FOVVPYVWWHUD4VY/bundle.json","state_url":"https://pith.science/pith/5ZJONB4OJL3FOVVPYVWWHUD4VY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5ZJONB4OJL3FOVVPYVWWHUD4VY/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":"2026-06-26T03:00:21Z","links":{"resolver":"https://pith.science/pith/5ZJONB4OJL3FOVVPYVWWHUD4VY","bundle":"https://pith.science/pith/5ZJONB4OJL3FOVVPYVWWHUD4VY/bundle.json","state":"https://pith.science/pith/5ZJONB4OJL3FOVVPYVWWHUD4VY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5ZJONB4OJL3FOVVPYVWWHUD4VY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:5ZJONB4OJL3FOVVPYVWWHUD4VY","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":"33aba11b2b12b4666bcdf28af933f5bd7cabd76b8bba4864715ee5165fdc059c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2009-03-27T16:26:46Z","title_canon_sha256":"1149651d7c13a2783e26aeb19301e12d7bbffdc8346cf13ca1cebfd81b66731f"},"schema_version":"1.0","source":{"id":"0903.4842","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0903.4842","created_at":"2026-05-18T04:12:10Z"},{"alias_kind":"arxiv_version","alias_value":"0903.4842v1","created_at":"2026-05-18T04:12:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0903.4842","created_at":"2026-05-18T04:12:10Z"},{"alias_kind":"pith_short_12","alias_value":"5ZJONB4OJL3F","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"5ZJONB4OJL3FOVVP","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"5ZJONB4O","created_at":"2026-05-18T12:25:58Z"}],"graph_snapshots":[{"event_id":"sha256:c5f0bc5c42b39fd20ac32bb8f2ecd1d2e53378ee2cd1208da004ea9e1be5cd45","target":"graph","created_at":"2026-05-18T04:12:10Z","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":"In connection with an unsolved problem of Bang (1951) we give a lower bound for the sum of the base volumes of cylinders covering a d-dimensional convex body in terms of the relevant basic measures of the given convex body. As an application we establish lower bounds on the number of k-dimensional flats (i.e. translates of k-dimensional linear subspaces) needed to cover all the integer points of a given convex body in d-dimensional Euclidean space for 0<k<d.","authors_text":"Alexander Litvak, Karoly Bezdek","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2009-03-27T16:26:46Z","title":"Covering convex bodies by cylinders and lattice points by flats"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0903.4842","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:cf7ea8d6e84fff6062edd2206f4095b8a455451617929561bb68b628123a5de5","target":"record","created_at":"2026-05-18T04:12:10Z","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":"33aba11b2b12b4666bcdf28af933f5bd7cabd76b8bba4864715ee5165fdc059c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2009-03-27T16:26:46Z","title_canon_sha256":"1149651d7c13a2783e26aeb19301e12d7bbffdc8346cf13ca1cebfd81b66731f"},"schema_version":"1.0","source":{"id":"0903.4842","kind":"arxiv","version":1}},"canonical_sha256":"ee52e6878e4af65756afc56d63d07cae0a7b3b095f5275997489f078c7e7ca65","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ee52e6878e4af65756afc56d63d07cae0a7b3b095f5275997489f078c7e7ca65","first_computed_at":"2026-05-18T04:12:10.730995Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:12:10.730995Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"z2ne3S3iHpfYVcpC1Y6xbC822s+Y0vd0J/3VKT5VM8IjX1ptcWvvgkvaTDes665M89qeaaCl3r6Q9FrhcZcgAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:12:10.731556Z","signed_message":"canonical_sha256_bytes"},"source_id":"0903.4842","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cf7ea8d6e84fff6062edd2206f4095b8a455451617929561bb68b628123a5de5","sha256:c5f0bc5c42b39fd20ac32bb8f2ecd1d2e53378ee2cd1208da004ea9e1be5cd45"],"state_sha256":"293b188f650e4250c36d18bff31e56c7cda2b00cbb31f7e848ec36f3dbc66905"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Bb9yi3FK9waCr7wBQreklPJjubOxfOLGCwVKzWmAtCbyMsyRXb5LrDSaPt36qnIugvzhvm0Tq6yQG3O/0aLgCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T03:00:21.750793Z","bundle_sha256":"6deb70c55bffc1c7a364fcde077626f276a8167edd1b6c0e18fab033ba56de60"}}