{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2009:6O75NLP3YMKNTOIUXBLBV3JIGJ","short_pith_number":"pith:6O75NLP3","schema_version":"1.0","canonical_sha256":"f3bfd6adfbc314d9b914b8561aed28326ea00da3d4db22b9da88c56869c46231","source":{"kind":"arxiv","id":"0908.1586","version":2},"attestation_state":"computed","paper":{"title":"Minimal half-spaces and external representation of tropical polyhedra","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.OC","authors_text":"Ricardo D. Katz, Stephane Gaubert","submitted_at":"2009-08-12T07:55:12Z","abstract_excerpt":"We give a characterization of the minimal tropical half-spaces containing a given tropical polyhedron, from which we derive a counter example showing that the number of such minimal half-spaces can be infinite, contradicting some statements which appeared in the tropical literature, and disproving a conjecture of F. Block and J. Yu. We also establish an analogue of the Minkowski-Weyl theorem, showing that a tropical polyhedron can be equivalently represented internally (in terms of extreme points and rays) or externally (in terms of half-spaces containing it). A canonical external representati"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"0908.1586","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2009-08-12T07:55:12Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"122c06762b97c4c7b308c902a9fa4b97d0391cbb899b3e385cd748f8231d85e0","abstract_canon_sha256":"edcc0f74a418638e933418dd425a10422d45fb31369b13bdf230ec48a6da4d3d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:24:43.963356Z","signature_b64":"fXAxuGL0V5Rtm9OVns/QkAvRqVTkHAMUvBAx798GMjy3AinEXFOqrdT2fg53i5ZqWJEs2OFo4Msrjo8I5AaYAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f3bfd6adfbc314d9b914b8561aed28326ea00da3d4db22b9da88c56869c46231","last_reissued_at":"2026-05-18T04:24:43.962989Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:24:43.962989Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Minimal half-spaces and external representation of tropical polyhedra","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.OC","authors_text":"Ricardo D. Katz, Stephane Gaubert","submitted_at":"2009-08-12T07:55:12Z","abstract_excerpt":"We give a characterization of the minimal tropical half-spaces containing a given tropical polyhedron, from which we derive a counter example showing that the number of such minimal half-spaces can be infinite, contradicting some statements which appeared in the tropical literature, and disproving a conjecture of F. Block and J. Yu. We also establish an analogue of the Minkowski-Weyl theorem, showing that a tropical polyhedron can be equivalently represented internally (in terms of extreme points and rays) or externally (in terms of half-spaces containing it). A canonical external representati"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0908.1586","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"0908.1586","created_at":"2026-05-18T04:24:43.963045+00:00"},{"alias_kind":"arxiv_version","alias_value":"0908.1586v2","created_at":"2026-05-18T04:24:43.963045+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0908.1586","created_at":"2026-05-18T04:24:43.963045+00:00"},{"alias_kind":"pith_short_12","alias_value":"6O75NLP3YMKN","created_at":"2026-05-18T12:25:58.837520+00:00"},{"alias_kind":"pith_short_16","alias_value":"6O75NLP3YMKNTOIU","created_at":"2026-05-18T12:25:58.837520+00:00"},{"alias_kind":"pith_short_8","alias_value":"6O75NLP3","created_at":"2026-05-18T12:25:58.837520+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/6O75NLP3YMKNTOIUXBLBV3JIGJ","json":"https://pith.science/pith/6O75NLP3YMKNTOIUXBLBV3JIGJ.json","graph_json":"https://pith.science/api/pith-number/6O75NLP3YMKNTOIUXBLBV3JIGJ/graph.json","events_json":"https://pith.science/api/pith-number/6O75NLP3YMKNTOIUXBLBV3JIGJ/events.json","paper":"https://pith.science/paper/6O75NLP3"},"agent_actions":{"view_html":"https://pith.science/pith/6O75NLP3YMKNTOIUXBLBV3JIGJ","download_json":"https://pith.science/pith/6O75NLP3YMKNTOIUXBLBV3JIGJ.json","view_paper":"https://pith.science/paper/6O75NLP3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0908.1586&json=true","fetch_graph":"https://pith.science/api/pith-number/6O75NLP3YMKNTOIUXBLBV3JIGJ/graph.json","fetch_events":"https://pith.science/api/pith-number/6O75NLP3YMKNTOIUXBLBV3JIGJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6O75NLP3YMKNTOIUXBLBV3JIGJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6O75NLP3YMKNTOIUXBLBV3JIGJ/action/storage_attestation","attest_author":"https://pith.science/pith/6O75NLP3YMKNTOIUXBLBV3JIGJ/action/author_attestation","sign_citation":"https://pith.science/pith/6O75NLP3YMKNTOIUXBLBV3JIGJ/action/citation_signature","submit_replication":"https://pith.science/pith/6O75NLP3YMKNTOIUXBLBV3JIGJ/action/replication_record"}},"created_at":"2026-05-18T04:24:43.963045+00:00","updated_at":"2026-05-18T04:24:43.963045+00:00"}