{"paper":{"title":"On mean outer radii of random polytopes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"David Alonso-Gutierrez, Joscha Prochno, Maria A. Hernandez Cifre, Nikos Dafnis","submitted_at":"2012-11-10T16:35:35Z","abstract_excerpt":"In this paper we introduce a new sequence of quantities for random polytopes. Let $K_N=\\conv\\{X_1,...,X_N\\}$ be a random polytope generated by independent random vectors uniformly distributed in an isotropic convex body $K$ of $\\R^n$. We prove that the so-called $k$-th mean outer radius $\\widetilde R_k(K_N)$ has order $\\max\\{\\sqrt{k},\\sqrt{\\log N}\\}L_K$ with high probability if $n^2\\leq N\\leq e^{\\sqrt{n}}$. We also show that this is also the right order of the expected value of $\\widetilde R_k(K_N)$ in the full range $n\\leq N\\leq e^{\\sqrt{n}}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.2336","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":""},"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"}