{"paper":{"title":"Estimating the probability that a given vector is in the convex hull of a random sample","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.ST","stat.TH"],"primary_cat":"math.PR","authors_text":"Harald Oberhauser, Satoshi Hayakawa, Terry Lyons","submitted_at":"2021-01-12T01:00:21Z","abstract_excerpt":"For a $d$-dimensional random vector $X$, let $p_{n, X}(\\theta)$ be the probability that the convex hull of $n$ independent copies of $X$ contains a given point $\\theta$. We provide several sharp inequalities regarding $p_{n, X}(\\theta)$ and $N_X(\\theta)$ denoting the smallest $n$ for which $p_{n, X}(\\theta)\\ge1/2$. As a main result, we derive the totally general inequality $1/2 \\le \\alpha_X(\\theta)N_X(\\theta)\\le 3d + 1$, where $\\alpha_X(\\theta)$ (a.k.a. the Tukey depth) is the minimum probability that $X$ is in a fixed closed halfspace containing the point $\\theta$. We also show several applic"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2101.04250","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/2101.04250/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"}