{"paper":{"title":"The relaxation complexity of the standard simplex is logarithmic","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO","math.OC"],"primary_cat":"cs.DM","authors_text":"Simon Keil, Stefan Weltge","submitted_at":"2026-06-10T09:30:03Z","abstract_excerpt":"For a set $X$ of integer points, the relaxation complexity $\\operatorname{rc}(X)$ is the smallest number of facets of any polyhedron $P$ such that $P \\cap \\mathbb{Z}^d = X$. In this paper, we focus on the case where $X$ is the discrete standard simplex $\\Delta_d = \\{\\mathbf{0}, \\mathbf{e}_1, \\dots, \\mathbf{e}_d\\}$. We show that $\\operatorname{rc}(\\Delta_d) = O(\\log d)$ by an explicit, elementary construction. This improves upon the previously best-known upper bound $\\operatorname{rc}(\\Delta_d) = O(d / \\sqrt{\\log d})$ due to Aprile, Averkov, Di Summa, and Hojny (2024) and matches an asymptotic "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.11852","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/2606.11852/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"}