{"paper":{"title":"Convex polytopes in restricted point sets in $\\mathbb{R}^d$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CO","authors_text":"Boris Bukh, Zichao Dong","submitted_at":"2022-04-05T20:57:22Z","abstract_excerpt":"For a finite point set $P \\subset \\mathbb{R}^d$, denote by $\\text{diam}(P)$ the ratio of the largest to the smallest distances between pairs of points in $P$. Let $c_{d, \\alpha}(n)$ be the largest integer $c$ such that any $n$-point set $P \\subset \\mathbb{R}^d$ in general position, satisfying $\\text{diam}(P) < \\alpha\\sqrt[d]{n}$, contains an $c$-point convex independent subset. We determine the asymptotics of $c_{d, \\alpha}(n)$ as $n \\to \\infty$ by showing the existence of positive constants $\\beta = \\beta(d, \\alpha)$ and $\\gamma = \\gamma(d)$ such that $\\beta n^{\\frac{d-1}{d+1}} \\le c_{d, \\alp"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2204.02487","kind":"arxiv","version":4},"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/2204.02487/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"}