{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:GA5TTJSOX2EVAZMN657VVLHBV5","short_pith_number":"pith:GA5TTJSO","canonical_record":{"source":{"id":"1202.5346","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-02-23T23:50:34Z","cross_cats_sorted":["math.GN"],"title_canon_sha256":"e197bf021e4e6e84e78db80ff2588cf8cd1f602d006caf69649c6939eb48dc37","abstract_canon_sha256":"fa590dcf3dce2c4a6394f977633012dcfbc7771df7067565554c60744b7da429"},"schema_version":"1.0"},"canonical_sha256":"303b39a64ebe8950658df77f5aace1af41d59a51f230007daa4ce7243702a7d2","source":{"kind":"arxiv","id":"1202.5346","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1202.5346","created_at":"2026-05-18T03:38:14Z"},{"alias_kind":"arxiv_version","alias_value":"1202.5346v1","created_at":"2026-05-18T03:38:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.5346","created_at":"2026-05-18T03:38:14Z"},{"alias_kind":"pith_short_12","alias_value":"GA5TTJSOX2EV","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_16","alias_value":"GA5TTJSOX2EVAZMN","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_8","alias_value":"GA5TTJSO","created_at":"2026-05-18T12:27:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:GA5TTJSOX2EVAZMN657VVLHBV5","target":"record","payload":{"canonical_record":{"source":{"id":"1202.5346","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-02-23T23:50:34Z","cross_cats_sorted":["math.GN"],"title_canon_sha256":"e197bf021e4e6e84e78db80ff2588cf8cd1f602d006caf69649c6939eb48dc37","abstract_canon_sha256":"fa590dcf3dce2c4a6394f977633012dcfbc7771df7067565554c60744b7da429"},"schema_version":"1.0"},"canonical_sha256":"303b39a64ebe8950658df77f5aace1af41d59a51f230007daa4ce7243702a7d2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:38:14.284026Z","signature_b64":"jEKovUdbYG4RylAZh2wbqzRIddbuFmmzd/6YFFHwwi/WkTfGNn9YGJj/Dk19UTzvpffN6gg2LaxIrcR01IoFBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"303b39a64ebe8950658df77f5aace1af41d59a51f230007daa4ce7243702a7d2","last_reissued_at":"2026-05-18T03:38:14.283555Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:38:14.283555Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1202.5346","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":"2026-05-18T03:38:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oIdvIHLumonziI6a5OiKCkWnNhVgPISMMgfnMyUswCon14i9RtCKfj9QwwOm6jNget+w7VwDkjM8aya6oQMKAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T06:47:10.978215Z"},"content_sha256":"85fe5eb78694d8cf90bbb6e82badda324b9f37fb45c06b6c728443f0349d8da6","schema_version":"1.0","event_id":"sha256:85fe5eb78694d8cf90bbb6e82badda324b9f37fb45c06b6c728443f0349d8da6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:GA5TTJSOX2EVAZMN657VVLHBV5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Compactly convex sets in linear topological spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GN"],"primary_cat":"math.FA","authors_text":"M. Mitrofanov, O. Ravsky, T. Banakh","submitted_at":"2012-02-23T23:50:34Z","abstract_excerpt":"A convex subset X of a linear topological space is called compactly convex if there is a continuous compact-valued map $\\Phi:X\\to exp(X)$ such that $[x,y]\\subset\\Phi(x)\\cup \\Phi(y)$ for all $x,y\\in X$. We prove that each convex subset of the plane is compactly convex. On the other hand, the space $R^3$ contains a convex set that is not compactly convex. Each compactly convex subset $X$ of a linear topological space $L$ has locally compact closure $\\bar X$ which is metrizable if and only if each compact subset of $X$ is metrizable."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.5346","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":"2026-05-18T03:38:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"19JIjRRk+4Uq+oiXhTwxJom4QV24bBOPPdCHxB2oczef7oMemQJba0Np/P+0yE26sXk5JEoMlT3CmZ03FcZWAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T06:47:10.978582Z"},"content_sha256":"4c4b804d0bfb7d194c4249ebbf7504706089f3ef57eb516138e5b4e81a726019","schema_version":"1.0","event_id":"sha256:4c4b804d0bfb7d194c4249ebbf7504706089f3ef57eb516138e5b4e81a726019"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GA5TTJSOX2EVAZMN657VVLHBV5/bundle.json","state_url":"https://pith.science/pith/GA5TTJSOX2EVAZMN657VVLHBV5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GA5TTJSOX2EVAZMN657VVLHBV5/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":"2026-06-23T06:47:10Z","links":{"resolver":"https://pith.science/pith/GA5TTJSOX2EVAZMN657VVLHBV5","bundle":"https://pith.science/pith/GA5TTJSOX2EVAZMN657VVLHBV5/bundle.json","state":"https://pith.science/pith/GA5TTJSOX2EVAZMN657VVLHBV5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GA5TTJSOX2EVAZMN657VVLHBV5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:GA5TTJSOX2EVAZMN657VVLHBV5","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":"fa590dcf3dce2c4a6394f977633012dcfbc7771df7067565554c60744b7da429","cross_cats_sorted":["math.GN"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-02-23T23:50:34Z","title_canon_sha256":"e197bf021e4e6e84e78db80ff2588cf8cd1f602d006caf69649c6939eb48dc37"},"schema_version":"1.0","source":{"id":"1202.5346","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1202.5346","created_at":"2026-05-18T03:38:14Z"},{"alias_kind":"arxiv_version","alias_value":"1202.5346v1","created_at":"2026-05-18T03:38:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.5346","created_at":"2026-05-18T03:38:14Z"},{"alias_kind":"pith_short_12","alias_value":"GA5TTJSOX2EV","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_16","alias_value":"GA5TTJSOX2EVAZMN","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_8","alias_value":"GA5TTJSO","created_at":"2026-05-18T12:27:06Z"}],"graph_snapshots":[{"event_id":"sha256:4c4b804d0bfb7d194c4249ebbf7504706089f3ef57eb516138e5b4e81a726019","target":"graph","created_at":"2026-05-18T03:38:14Z","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":"A convex subset X of a linear topological space is called compactly convex if there is a continuous compact-valued map $\\Phi:X\\to exp(X)$ such that $[x,y]\\subset\\Phi(x)\\cup \\Phi(y)$ for all $x,y\\in X$. We prove that each convex subset of the plane is compactly convex. On the other hand, the space $R^3$ contains a convex set that is not compactly convex. Each compactly convex subset $X$ of a linear topological space $L$ has locally compact closure $\\bar X$ which is metrizable if and only if each compact subset of $X$ is metrizable.","authors_text":"M. Mitrofanov, O. Ravsky, T. Banakh","cross_cats":["math.GN"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-02-23T23:50:34Z","title":"Compactly convex sets in linear topological spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.5346","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":"sha256:85fe5eb78694d8cf90bbb6e82badda324b9f37fb45c06b6c728443f0349d8da6","target":"record","created_at":"2026-05-18T03:38:14Z","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":"fa590dcf3dce2c4a6394f977633012dcfbc7771df7067565554c60744b7da429","cross_cats_sorted":["math.GN"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-02-23T23:50:34Z","title_canon_sha256":"e197bf021e4e6e84e78db80ff2588cf8cd1f602d006caf69649c6939eb48dc37"},"schema_version":"1.0","source":{"id":"1202.5346","kind":"arxiv","version":1}},"canonical_sha256":"303b39a64ebe8950658df77f5aace1af41d59a51f230007daa4ce7243702a7d2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"303b39a64ebe8950658df77f5aace1af41d59a51f230007daa4ce7243702a7d2","first_computed_at":"2026-05-18T03:38:14.283555Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:38:14.283555Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jEKovUdbYG4RylAZh2wbqzRIddbuFmmzd/6YFFHwwi/WkTfGNn9YGJj/Dk19UTzvpffN6gg2LaxIrcR01IoFBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:38:14.284026Z","signed_message":"canonical_sha256_bytes"},"source_id":"1202.5346","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:85fe5eb78694d8cf90bbb6e82badda324b9f37fb45c06b6c728443f0349d8da6","sha256:4c4b804d0bfb7d194c4249ebbf7504706089f3ef57eb516138e5b4e81a726019"],"state_sha256":"ff3df0acf9148679b732fd50910c79a222895f289372909e2b63911573f30b87"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WHNzVGldMLVZX3LZFQiUpw5QSuH1QG5gWFT/zeQ1L9iUZ7FdRei8Si5jCKKzDa0clDJHOwxigdXnqNo18s2rAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T06:47:10.980467Z","bundle_sha256":"f333a32ae0efe17d7b7dcf5975abd9209bd2cb265a5c0fb2714ca143ef98d173"}}