{"paper":{"title":"Geometry of a Set and its Random covers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.PR","authors_text":"Bala Krishnamoorthy, Enrique Alvarado, Kevin R. Vixie","submitted_at":"2021-12-30T09:28:59Z","abstract_excerpt":"Let $E$ be a bounded open subset of $\\mathbb{R}^n$. We study the following questions: For i.i.d. samples $X_1, \\dots, X_N$ drawn uniformly from $E$, what is the probability that $\\cup_i \\mathbf{B}(X_i, \\delta)$, the union of $\\delta$-balls centered at $X_i$, covers $E$? And how does the probability depend on sample size $N$ and the radius of balls $\\delta$? We present geometric conditions of $E$ under which we derive lower bounds to this probability. These lower bounds tend to $1$ as a function of $\\exp{(-\\delta^n N)}$.\n  The basic tool that we use to derive the lower bounds is a good partitio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2112.14979","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/2112.14979/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"}