{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:RBZLOKNKL44FE64OQFQOIO3AGI","short_pith_number":"pith:RBZLOKNK","canonical_record":{"source":{"id":"1111.6708","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-11-29T07:01:34Z","cross_cats_sorted":["math.GN"],"title_canon_sha256":"71f7054833ad0629a10d4b35026d426ae1eeb378f0a73cde8fb35bb6df9c9116","abstract_canon_sha256":"8ed6b85bb2e461efb4b4ea9f8b0893c71b4256981177f42a985e6dda20efebc6"},"schema_version":"1.0"},"canonical_sha256":"8872b729aa5f38527b8e8160e43b60323ef3b05a8de833f79c6391de7e0b229d","source":{"kind":"arxiv","id":"1111.6708","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.6708","created_at":"2026-05-18T03:38:14Z"},{"alias_kind":"arxiv_version","alias_value":"1111.6708v1","created_at":"2026-05-18T03:38:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.6708","created_at":"2026-05-18T03:38:14Z"},{"alias_kind":"pith_short_12","alias_value":"RBZLOKNKL44F","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_16","alias_value":"RBZLOKNKL44FE64O","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_8","alias_value":"RBZLOKNK","created_at":"2026-05-18T12:26:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:RBZLOKNKL44FE64OQFQOIO3AGI","target":"record","payload":{"canonical_record":{"source":{"id":"1111.6708","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-11-29T07:01:34Z","cross_cats_sorted":["math.GN"],"title_canon_sha256":"71f7054833ad0629a10d4b35026d426ae1eeb378f0a73cde8fb35bb6df9c9116","abstract_canon_sha256":"8ed6b85bb2e461efb4b4ea9f8b0893c71b4256981177f42a985e6dda20efebc6"},"schema_version":"1.0"},"canonical_sha256":"8872b729aa5f38527b8e8160e43b60323ef3b05a8de833f79c6391de7e0b229d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:38:14.342138Z","signature_b64":"PSASX9cKjWZPYHVYddNm9kfwS6buSaUv67ADmH31MUL10BYfCro74O3Rx8OSkk4CJBarDRnaSPs6xHrua8c5Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8872b729aa5f38527b8e8160e43b60323ef3b05a8de833f79c6391de7e0b229d","last_reissued_at":"2026-05-18T03:38:14.341547Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:38:14.341547Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1111.6708","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":"FUIKTQeVFKsE1dKNnAmZzPZhsC0GYD6jxG4snss632EIe3+RopVg6lG5Zo28xNmZWgiqKEtTGj4GH7rWeMrtBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-22T13:56:18.806482Z"},"content_sha256":"8911803bab2911fa897dfb242522a90fc279ca4d893d3095c991d7928c55117a","schema_version":"1.0","event_id":"sha256:8911803bab2911fa897dfb242522a90fc279ca4d893d3095c991d7928c55117a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:RBZLOKNKL44FE64OQFQOIO3AGI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A \"hidden\" characterization of approximatively polyhedral convex sets in Banach spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GN"],"primary_cat":"math.FA","authors_text":"Ivan Hetman, Taras Banakh","submitted_at":"2011-11-29T07:01:34Z","abstract_excerpt":"For a Banach space $X$ by $Conv_H(X)$ we denote the space of non-empty closed convex subsets of $X$, endowed with the Hausdorff metric. We prove that for any closed convex set $C\\subset X$ and its metric component $H_C=\\{A\\in Conv_H(X):d_H(A,C)<\\infty\\}$ in $Conv_H(X)$, the following conditions are equivalent: (1) $C$ is approximatively polyhedral, which means that for every $\\epsilon>0$ there is a polyhedral convex subset $P\\subset X$ on Hausdorff distance $d_H(P,C)<\\epsilon$ from $C$; (2) $C$ lies on finite Hausdorff distance $d_H(C,P)$ from some polyhedral convex set $P\\subset X$; (3) the m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.6708","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":"wbTqtYxNigLQNS/2v6m6vzT379IDmcib6876kPTODlwy8vQDDhu/2USle8AFw3hbzK4QYxcjgmwMTsSmsFA5BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-22T13:56:18.807188Z"},"content_sha256":"9b8cd3868996b6d89f6ef733732c7554db5da062246746386f07ed716524f893","schema_version":"1.0","event_id":"sha256:9b8cd3868996b6d89f6ef733732c7554db5da062246746386f07ed716524f893"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RBZLOKNKL44FE64OQFQOIO3AGI/bundle.json","state_url":"https://pith.science/pith/RBZLOKNKL44FE64OQFQOIO3AGI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RBZLOKNKL44FE64OQFQOIO3AGI/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":"2026-05-22T13:56:18Z","links":{"resolver":"https://pith.science/pith/RBZLOKNKL44FE64OQFQOIO3AGI","bundle":"https://pith.science/pith/RBZLOKNKL44FE64OQFQOIO3AGI/bundle.json","state":"https://pith.science/pith/RBZLOKNKL44FE64OQFQOIO3AGI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RBZLOKNKL44FE64OQFQOIO3AGI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:RBZLOKNKL44FE64OQFQOIO3AGI","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":"8ed6b85bb2e461efb4b4ea9f8b0893c71b4256981177f42a985e6dda20efebc6","cross_cats_sorted":["math.GN"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-11-29T07:01:34Z","title_canon_sha256":"71f7054833ad0629a10d4b35026d426ae1eeb378f0a73cde8fb35bb6df9c9116"},"schema_version":"1.0","source":{"id":"1111.6708","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.6708","created_at":"2026-05-18T03:38:14Z"},{"alias_kind":"arxiv_version","alias_value":"1111.6708v1","created_at":"2026-05-18T03:38:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.6708","created_at":"2026-05-18T03:38:14Z"},{"alias_kind":"pith_short_12","alias_value":"RBZLOKNKL44F","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_16","alias_value":"RBZLOKNKL44FE64O","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_8","alias_value":"RBZLOKNK","created_at":"2026-05-18T12:26:41Z"}],"graph_snapshots":[{"event_id":"sha256:9b8cd3868996b6d89f6ef733732c7554db5da062246746386f07ed716524f893","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":"For a Banach space $X$ by $Conv_H(X)$ we denote the space of non-empty closed convex subsets of $X$, endowed with the Hausdorff metric. We prove that for any closed convex set $C\\subset X$ and its metric component $H_C=\\{A\\in Conv_H(X):d_H(A,C)<\\infty\\}$ in $Conv_H(X)$, the following conditions are equivalent: (1) $C$ is approximatively polyhedral, which means that for every $\\epsilon>0$ there is a polyhedral convex subset $P\\subset X$ on Hausdorff distance $d_H(P,C)<\\epsilon$ from $C$; (2) $C$ lies on finite Hausdorff distance $d_H(C,P)$ from some polyhedral convex set $P\\subset X$; (3) the m","authors_text":"Ivan Hetman, Taras Banakh","cross_cats":["math.GN"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-11-29T07:01:34Z","title":"A \"hidden\" characterization of approximatively polyhedral convex sets in Banach spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.6708","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":"sha256:8911803bab2911fa897dfb242522a90fc279ca4d893d3095c991d7928c55117a","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":"8ed6b85bb2e461efb4b4ea9f8b0893c71b4256981177f42a985e6dda20efebc6","cross_cats_sorted":["math.GN"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-11-29T07:01:34Z","title_canon_sha256":"71f7054833ad0629a10d4b35026d426ae1eeb378f0a73cde8fb35bb6df9c9116"},"schema_version":"1.0","source":{"id":"1111.6708","kind":"arxiv","version":1}},"canonical_sha256":"8872b729aa5f38527b8e8160e43b60323ef3b05a8de833f79c6391de7e0b229d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8872b729aa5f38527b8e8160e43b60323ef3b05a8de833f79c6391de7e0b229d","first_computed_at":"2026-05-18T03:38:14.341547Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:38:14.341547Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"PSASX9cKjWZPYHVYddNm9kfwS6buSaUv67ADmH31MUL10BYfCro74O3Rx8OSkk4CJBarDRnaSPs6xHrua8c5Dg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:38:14.342138Z","signed_message":"canonical_sha256_bytes"},"source_id":"1111.6708","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8911803bab2911fa897dfb242522a90fc279ca4d893d3095c991d7928c55117a","sha256:9b8cd3868996b6d89f6ef733732c7554db5da062246746386f07ed716524f893"],"state_sha256":"187b50115cf694eea140653b1616ba849fbb574118816c25a1cfda832858277b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NDwpfwNVhPb2RjT1YAcseVXHlm7VE/GwZc6RTE8KJDVlaflE+OvXiz+YRzjvB3C+gHSFaNpIP0tdv7Rd/TRWAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-22T13:56:18.811993Z","bundle_sha256":"c6261619e84b0f6826b929cbad86a8b934cb5b4762204276ca0fda4f5ecf9d56"}}