{"paper":{"title":"The covering radius of randomly distributed points on a manifold","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"A. Reznikov, E. B. Saff","submitted_at":"2015-04-12T22:18:24Z","abstract_excerpt":"We derive fundamental asymptotic results for the expected covering radius $\\rho(X_N)$ for $N$ points that are randomly and independently distributed with respect to surface measure on a sphere as well as on a class of smooth manifolds. For the unit sphere $\\mathbb{S}^d \\subset \\mathbb{R}^{d+1}$, we obtain the precise asymptotic that $\\mathbb{E}\\rho(X_N)[N/\\log N]^{1/d}$ has limit $[(d+1)\\upsilon_{d+1}/\\upsilon_d]^{1/d}$ as $N \\to \\infty $, where $\\upsilon_d$ is the volume of the $d$-dimensional unit ball. This proves a recent conjecture of Brauchart et al. as well as extends a result previousl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.03029","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"}