{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:ABBYLNGXJPNU2WUNEYV6L5MF23","short_pith_number":"pith:ABBYLNGX","schema_version":"1.0","canonical_sha256":"004385b4d74bdb4d5a8d262be5f585d6f09248c45a39b58cc7e599ccda3ea730","source":{"kind":"arxiv","id":"1611.07826","version":2},"attestation_state":"computed","paper":{"title":"A generalization of the concept of distance based on the simplex inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.MG","authors_text":"Bruno Teheux, Gergely Kiss, Jean-Luc Marichal","submitted_at":"2016-11-23T17:54:40Z","abstract_excerpt":"We introduce and discuss the concept of \\emph{$n$-distance}, a generalization to $n$ elements of the classical notion of distance obtained by replacing the triangle inequality with the so-called simplex inequality $$ d(x_1, \\ldots, x_n)~\\leq~K\\, \\sum_{i=1}^n d(x_1, \\ldots, x_n)_i^z{\\,}, \\qquad x_1, \\ldots, x_n, z \\in X, $$ where $K=1$. Here $d(x_1,\\ldots,x_n)_i^z$ is obtained from the function $d(x_1,\\ldots,x_n)$ by setting its $i$th variable to $z$. We provide several examples of $n$-distances, and for each of them we investigate the infimum of the set of real numbers $K\\in\\left]0,1\\right]$ f"},"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":"1611.07826","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-11-23T17:54:40Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"9a0f956ec0099b4508db270c6e50842b65c5f1d95acede835ffe5603b96c7a98","abstract_canon_sha256":"41e5f4382daa1888b016671a974e4f63031e5f808407a2b937949ccc69802656"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:15:30.997734Z","signature_b64":"PqAcSvrmv0IlwMe3RMc8qyJ3ptXkisHbN/FtgMRLjvYGK/hNe4YiwxKKdo0x1nOZ2rQmrV9di3wVDJdN+65oBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"004385b4d74bdb4d5a8d262be5f585d6f09248c45a39b58cc7e599ccda3ea730","last_reissued_at":"2026-05-18T00:15:30.997107Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:15:30.997107Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A generalization of the concept of distance based on the simplex inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.MG","authors_text":"Bruno Teheux, Gergely Kiss, Jean-Luc Marichal","submitted_at":"2016-11-23T17:54:40Z","abstract_excerpt":"We introduce and discuss the concept of \\emph{$n$-distance}, a generalization to $n$ elements of the classical notion of distance obtained by replacing the triangle inequality with the so-called simplex inequality $$ d(x_1, \\ldots, x_n)~\\leq~K\\, \\sum_{i=1}^n d(x_1, \\ldots, x_n)_i^z{\\,}, \\qquad x_1, \\ldots, x_n, z \\in X, $$ where $K=1$. Here $d(x_1,\\ldots,x_n)_i^z$ is obtained from the function $d(x_1,\\ldots,x_n)$ by setting its $i$th variable to $z$. We provide several examples of $n$-distances, and for each of them we investigate the infimum of the set of real numbers $K\\in\\left]0,1\\right]$ f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.07826","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":"1611.07826","created_at":"2026-05-18T00:15:30.997196+00:00"},{"alias_kind":"arxiv_version","alias_value":"1611.07826v2","created_at":"2026-05-18T00:15:30.997196+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.07826","created_at":"2026-05-18T00:15:30.997196+00:00"},{"alias_kind":"pith_short_12","alias_value":"ABBYLNGXJPNU","created_at":"2026-05-18T12:30:04.600751+00:00"},{"alias_kind":"pith_short_16","alias_value":"ABBYLNGXJPNU2WUN","created_at":"2026-05-18T12:30:04.600751+00:00"},{"alias_kind":"pith_short_8","alias_value":"ABBYLNGX","created_at":"2026-05-18T12:30:04.600751+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/ABBYLNGXJPNU2WUNEYV6L5MF23","json":"https://pith.science/pith/ABBYLNGXJPNU2WUNEYV6L5MF23.json","graph_json":"https://pith.science/api/pith-number/ABBYLNGXJPNU2WUNEYV6L5MF23/graph.json","events_json":"https://pith.science/api/pith-number/ABBYLNGXJPNU2WUNEYV6L5MF23/events.json","paper":"https://pith.science/paper/ABBYLNGX"},"agent_actions":{"view_html":"https://pith.science/pith/ABBYLNGXJPNU2WUNEYV6L5MF23","download_json":"https://pith.science/pith/ABBYLNGXJPNU2WUNEYV6L5MF23.json","view_paper":"https://pith.science/paper/ABBYLNGX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1611.07826&json=true","fetch_graph":"https://pith.science/api/pith-number/ABBYLNGXJPNU2WUNEYV6L5MF23/graph.json","fetch_events":"https://pith.science/api/pith-number/ABBYLNGXJPNU2WUNEYV6L5MF23/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ABBYLNGXJPNU2WUNEYV6L5MF23/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ABBYLNGXJPNU2WUNEYV6L5MF23/action/storage_attestation","attest_author":"https://pith.science/pith/ABBYLNGXJPNU2WUNEYV6L5MF23/action/author_attestation","sign_citation":"https://pith.science/pith/ABBYLNGXJPNU2WUNEYV6L5MF23/action/citation_signature","submit_replication":"https://pith.science/pith/ABBYLNGXJPNU2WUNEYV6L5MF23/action/replication_record"}},"created_at":"2026-05-18T00:15:30.997196+00:00","updated_at":"2026-05-18T00:15:30.997196+00:00"}