{"paper":{"title":"Expected intrinsic volumes and facet numbers of random beta-polytopes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.MG","authors_text":"Christoph Thaele, Daniel Temesvari, Zakhar Kabluchko","submitted_at":"2017-07-07T16:16:39Z","abstract_excerpt":"Let $X_1,\\ldots,X_n$ be i.i.d.\\ random points in the $d$-dimensional Euclidean space sampled according to one of the following probability densities: $$ f_{d,\\beta} (x) = \\text{const} \\cdot (1-\\|x\\|^2)^{\\beta}, \\quad \\|x\\|\\leq 1, \\quad \\text{(the beta case)} $$ and $$ \\tilde f_{d,\\beta} (x) = \\text{const} \\cdot (1+\\|x\\|^2)^{-\\beta}, \\quad x\\in\\mathbb{R}^d, \\quad \\text{(the beta' case).} $$ We compute exactly the expected intrinsic volumes and the expected number of facets of the convex hull of $X_1,\\ldots,X_n$. Asymptotic formulae where obtained previously by Affentranger [The convex hull of r"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.02253","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":""},"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"}