{"paper":{"title":"Probability that $n$ points are in convex position in a general convex polygon: Asymptotic results","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Ludovic Morin","submitted_at":"2024-10-15T15:44:32Z","abstract_excerpt":"Let $\\mathbb{P}_K(n)$ be the probability that $n$ points $z_1,\\ldots,z_n$ picked uniformly and independently in $K$, a non-flat compact convex polygon in $\\mathbb{R}^2$, are in convex position, that is, form the vertex set of a convex polygon. In this paper, we give an equivalent of $\\mathbb{P}_K(n)$ when $n\\to\\infty$. This improves on a famous result of B\\'ar\\'any (yet valid for a general convex domain $K$) and a result we initiated in the case where $K$ is a regular convex polygon."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2410.11706","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2410.11706/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"}