{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:FSHYKEEKHL4Y7ZMZING7JHFGUC","short_pith_number":"pith:FSHYKEEK","canonical_record":{"source":{"id":"1203.0578","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2012-03-02T21:44:59Z","cross_cats_sorted":["math.NA"],"title_canon_sha256":"29a77e1bdd4c0c5f171921826177bf808e821df89d7b18193bfbe0fe8e5b92b6","abstract_canon_sha256":"74ad67f2cafc332fa7233c83550974272f3d788b12358c8b958ba4a320a08741"},"schema_version":"1.0"},"canonical_sha256":"2c8f85108a3af98fe599434df49ca6a0a0128d2be172cb17d99618f49efe8313","source":{"kind":"arxiv","id":"1203.0578","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1203.0578","created_at":"2026-05-18T02:21:17Z"},{"alias_kind":"arxiv_version","alias_value":"1203.0578v2","created_at":"2026-05-18T02:21:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.0578","created_at":"2026-05-18T02:21:17Z"},{"alias_kind":"pith_short_12","alias_value":"FSHYKEEKHL4Y","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_16","alias_value":"FSHYKEEKHL4Y7ZMZ","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_8","alias_value":"FSHYKEEK","created_at":"2026-05-18T12:27:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:FSHYKEEKHL4Y7ZMZING7JHFGUC","target":"record","payload":{"canonical_record":{"source":{"id":"1203.0578","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2012-03-02T21:44:59Z","cross_cats_sorted":["math.NA"],"title_canon_sha256":"29a77e1bdd4c0c5f171921826177bf808e821df89d7b18193bfbe0fe8e5b92b6","abstract_canon_sha256":"74ad67f2cafc332fa7233c83550974272f3d788b12358c8b958ba4a320a08741"},"schema_version":"1.0"},"canonical_sha256":"2c8f85108a3af98fe599434df49ca6a0a0128d2be172cb17d99618f49efe8313","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:21:17.317663Z","signature_b64":"x4ioMPh2+I+4uCP7jkzBXYRV4z3JH2/D7SuHSPtZaUXAG5G1re7yMDgjyoJcnt0cXkdhWzogu+LWgPaPtmdmAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2c8f85108a3af98fe599434df49ca6a0a0128d2be172cb17d99618f49efe8313","last_reissued_at":"2026-05-18T02:21:17.316919Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:21:17.316919Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1203.0578","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":"2026-05-18T02:21:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OjFxz4PNYeD8mrF9iyxfUkcMUUHlKWytrXpfXYdT7w3ubLy/CdkMvFr4BZQEbvsrfRE10OJb83yB7FAx6VaCCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T06:25:44.293445Z"},"content_sha256":"cae67ee327daa8a60cfe562c1d06a2a239d3cf020fa2e10a72c02d85a18c237b","schema_version":"1.0","event_id":"sha256:cae67ee327daa8a60cfe562c1d06a2a239d3cf020fa2e10a72c02d85a18c237b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:FSHYKEEKHL4Y7ZMZING7JHFGUC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Look at the Generalized Heron Problem through the Lens of Majorization-Minimization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NA"],"primary_cat":"math.OC","authors_text":"Eric C. Chi, Kenneth Lange","submitted_at":"2012-03-02T21:44:59Z","abstract_excerpt":"In a recent issue of this journal, Mordukhovich et al.\\ pose and solve an interesting non-differentiable generalization of the Heron problem in the framework of modern convex analysis. In the generalized Heron problem one is given $k+1$ closed convex sets in $\\Real^d$ equipped with its Euclidean norm and asked to find the point in the last set such that the sum of the distances to the first $k$ sets is minimal. In later work the authors generalize the Heron problem even further, relax its convexity assumptions, study its theoretical properties, and pursue subgradient algorithms for solving the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.0578","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":"2026-05-18T02:21:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rE2WGCYcn4ExWuk4z/8O6d6Je7vNkkpR2wqbYNxiQX50L2rgbkZxYsy4h7ttxC6tcdtiW0yqdCqZphdUA64eBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T06:25:44.294213Z"},"content_sha256":"e5962a0bf517aad2b0f6458e8dbf0c72c69f1ad59403bddc8ff265892d6f298a","schema_version":"1.0","event_id":"sha256:e5962a0bf517aad2b0f6458e8dbf0c72c69f1ad59403bddc8ff265892d6f298a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FSHYKEEKHL4Y7ZMZING7JHFGUC/bundle.json","state_url":"https://pith.science/pith/FSHYKEEKHL4Y7ZMZING7JHFGUC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FSHYKEEKHL4Y7ZMZING7JHFGUC/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":"2026-06-01T06:25:44Z","links":{"resolver":"https://pith.science/pith/FSHYKEEKHL4Y7ZMZING7JHFGUC","bundle":"https://pith.science/pith/FSHYKEEKHL4Y7ZMZING7JHFGUC/bundle.json","state":"https://pith.science/pith/FSHYKEEKHL4Y7ZMZING7JHFGUC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FSHYKEEKHL4Y7ZMZING7JHFGUC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:FSHYKEEKHL4Y7ZMZING7JHFGUC","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":"74ad67f2cafc332fa7233c83550974272f3d788b12358c8b958ba4a320a08741","cross_cats_sorted":["math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2012-03-02T21:44:59Z","title_canon_sha256":"29a77e1bdd4c0c5f171921826177bf808e821df89d7b18193bfbe0fe8e5b92b6"},"schema_version":"1.0","source":{"id":"1203.0578","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1203.0578","created_at":"2026-05-18T02:21:17Z"},{"alias_kind":"arxiv_version","alias_value":"1203.0578v2","created_at":"2026-05-18T02:21:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.0578","created_at":"2026-05-18T02:21:17Z"},{"alias_kind":"pith_short_12","alias_value":"FSHYKEEKHL4Y","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_16","alias_value":"FSHYKEEKHL4Y7ZMZ","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_8","alias_value":"FSHYKEEK","created_at":"2026-05-18T12:27:06Z"}],"graph_snapshots":[{"event_id":"sha256:e5962a0bf517aad2b0f6458e8dbf0c72c69f1ad59403bddc8ff265892d6f298a","target":"graph","created_at":"2026-05-18T02:21:17Z","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 a recent issue of this journal, Mordukhovich et al.\\ pose and solve an interesting non-differentiable generalization of the Heron problem in the framework of modern convex analysis. In the generalized Heron problem one is given $k+1$ closed convex sets in $\\Real^d$ equipped with its Euclidean norm and asked to find the point in the last set such that the sum of the distances to the first $k$ sets is minimal. In later work the authors generalize the Heron problem even further, relax its convexity assumptions, study its theoretical properties, and pursue subgradient algorithms for solving the","authors_text":"Eric C. Chi, Kenneth Lange","cross_cats":["math.NA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2012-03-02T21:44:59Z","title":"A Look at the Generalized Heron Problem through the Lens of Majorization-Minimization"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.0578","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:cae67ee327daa8a60cfe562c1d06a2a239d3cf020fa2e10a72c02d85a18c237b","target":"record","created_at":"2026-05-18T02:21:17Z","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":"74ad67f2cafc332fa7233c83550974272f3d788b12358c8b958ba4a320a08741","cross_cats_sorted":["math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2012-03-02T21:44:59Z","title_canon_sha256":"29a77e1bdd4c0c5f171921826177bf808e821df89d7b18193bfbe0fe8e5b92b6"},"schema_version":"1.0","source":{"id":"1203.0578","kind":"arxiv","version":2}},"canonical_sha256":"2c8f85108a3af98fe599434df49ca6a0a0128d2be172cb17d99618f49efe8313","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2c8f85108a3af98fe599434df49ca6a0a0128d2be172cb17d99618f49efe8313","first_computed_at":"2026-05-18T02:21:17.316919Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:21:17.316919Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"x4ioMPh2+I+4uCP7jkzBXYRV4z3JH2/D7SuHSPtZaUXAG5G1re7yMDgjyoJcnt0cXkdhWzogu+LWgPaPtmdmAA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:21:17.317663Z","signed_message":"canonical_sha256_bytes"},"source_id":"1203.0578","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cae67ee327daa8a60cfe562c1d06a2a239d3cf020fa2e10a72c02d85a18c237b","sha256:e5962a0bf517aad2b0f6458e8dbf0c72c69f1ad59403bddc8ff265892d6f298a"],"state_sha256":"db86c6a310d5c025bb17841fba9a11823309dd79da9d247f2f607db0fc69c77e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GHFwiL9fVJ9dU2VQh1nd9ZkFHS32pyJHtAtSilanh6ieeyyhiI7H8qWLNP9xVS5qKSIKb+kk/Y2TJwTa1Z66Ag==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T06:25:44.298169Z","bundle_sha256":"160bb52e5fc65a6426da89afa0124aa26d81880f91236ca00d61d4bf1a7e0948"}}