{"paper":{"title":"Coverage processes on spheres and condition numbers for linear programming","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OC"],"primary_cat":"math.PR","authors_text":"Felipe Cucker, Martin Lotz, Peter B\\\"urgisser","submitted_at":"2007-12-17T20:58:49Z","abstract_excerpt":"This paper has two agendas. Firstly, we exhibit new results for coverage processes. Let $p(n,m,\\alpha)$ be the probability that $n$ spherical caps of angular radius $\\alpha$ in $S^m$ do not cover the whole sphere $S^m$. We give an exact formula for $p(n,m,\\alpha)$ in the case $\\alpha\\in[\\pi/2,\\pi]$ and an upper bound for $p(n,m,\\alpha)$ in the case $\\alpha\\in [0,\\pi/2]$ which tends to $p(n,m,\\pi/2)$ when $\\alpha\\to\\pi/2$. In the case $\\alpha\\in[0,\\pi/2]$ this yields upper bounds for the expected number of spherical caps of radius $\\alpha$ that are needed to cover $S^m$. Secondly, we study the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0712.2816","kind":"arxiv","version":3},"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"}