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Specifically, $ \\mathrm{width}(\\Delta_n) = \\sqrt{\\frac{2}{n + 1}}$ if $n$ is odd, and $ \\mathrm{width}(\\Delta_n) = \\sqrt{\\frac{2(n+1)}{n(n+2)}} $ if $n$ is even. While this bound is well known [GK92, Ale77], we provide a self-contained elementary proof that might (or might not) be of interest."},"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":"2301.02616","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2023-01-06T17:30:36Z","cross_cats_sorted":[],"title_canon_sha256":"c69a2a5911dcc5fa52289bb6603bf0e6d402d629b0416927b33e15ffad365bda","abstract_canon_sha256":"c14a5f9130a326fb0014c45ec3289232578799b18edc8ae7ff5df43fdb687b13"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:31:08.740206Z","signature_b64":"W0sZT9r+/9Gkfri55paKH5Rtci04k37/wns3+tHzUXicF6Y7+1aMUhFroW0ZnXW3rWQszMr5DdaPKt7j4uqiBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b444194c75b67a2570a5233635ec94bb75cf133fb34cbefa10e634a7d121f952","last_reissued_at":"2026-07-05T05:31:08.739872Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:31:08.739872Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Width of the Regular $n$-Simplex","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CG","authors_text":"Eliot W. Robson, Sariel Har-Peled","submitted_at":"2023-01-06T17:30:36Z","abstract_excerpt":"Consider the regular $n$-simplex $\\Delta_n$ - it is formed by the convex-hull of $n+1$ points in Euclidean space, with each pair of points being in distance exactly one from each other. We prove an exact bound on the width of $\\Delta_n$ which is $\\approx \\sqrt{2/n}$. Specifically, $ \\mathrm{width}(\\Delta_n) = \\sqrt{\\frac{2}{n + 1}}$ if $n$ is odd, and $ \\mathrm{width}(\\Delta_n) = \\sqrt{\\frac{2(n+1)}{n(n+2)}} $ if $n$ is even. While this bound is well known [GK92, Ale77], we provide a self-contained elementary proof that might (or might not) be of interest."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2301.02616","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2301.02616/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2301.02616","created_at":"2026-07-05T05:31:08.739927+00:00"},{"alias_kind":"arxiv_version","alias_value":"2301.02616v1","created_at":"2026-07-05T05:31:08.739927+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2301.02616","created_at":"2026-07-05T05:31:08.739927+00:00"},{"alias_kind":"pith_short_12","alias_value":"WRCBSTDVWZ5C","created_at":"2026-07-05T05:31:08.739927+00:00"},{"alias_kind":"pith_short_16","alias_value":"WRCBSTDVWZ5CK4FF","created_at":"2026-07-05T05:31:08.739927+00:00"},{"alias_kind":"pith_short_8","alias_value":"WRCBSTDV","created_at":"2026-07-05T05:31:08.739927+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/WRCBSTDVWZ5CK4FFEM3DL3EUXN","json":"https://pith.science/pith/WRCBSTDVWZ5CK4FFEM3DL3EUXN.json","graph_json":"https://pith.science/api/pith-number/WRCBSTDVWZ5CK4FFEM3DL3EUXN/graph.json","events_json":"https://pith.science/api/pith-number/WRCBSTDVWZ5CK4FFEM3DL3EUXN/events.json","paper":"https://pith.science/paper/WRCBSTDV"},"agent_actions":{"view_html":"https://pith.science/pith/WRCBSTDVWZ5CK4FFEM3DL3EUXN","download_json":"https://pith.science/pith/WRCBSTDVWZ5CK4FFEM3DL3EUXN.json","view_paper":"https://pith.science/paper/WRCBSTDV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2301.02616&json=true","fetch_graph":"https://pith.science/api/pith-number/WRCBSTDVWZ5CK4FFEM3DL3EUXN/graph.json","fetch_events":"https://pith.science/api/pith-number/WRCBSTDVWZ5CK4FFEM3DL3EUXN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WRCBSTDVWZ5CK4FFEM3DL3EUXN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WRCBSTDVWZ5CK4FFEM3DL3EUXN/action/storage_attestation","attest_author":"https://pith.science/pith/WRCBSTDVWZ5CK4FFEM3DL3EUXN/action/author_attestation","sign_citation":"https://pith.science/pith/WRCBSTDVWZ5CK4FFEM3DL3EUXN/action/citation_signature","submit_replication":"https://pith.science/pith/WRCBSTDVWZ5CK4FFEM3DL3EUXN/action/replication_record"}},"created_at":"2026-07-05T05:31:08.739927+00:00","updated_at":"2026-07-05T05:31:08.739927+00:00"}