{"paper":{"title":"Distance Functions, Critical Points, and the Topology of Random \\v{C}ech Complexes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.PR","authors_text":"Omer Bobrowski, Robert J. Adler","submitted_at":"2011-07-24T17:39:13Z","abstract_excerpt":"For a finite set of points $P$ in $R^d$, the function $d_P: R^d \\to R^+$ measures Euclidean distance to the set $P$. We study the number of critical points of $d_P$ when $P$ is a Poisson process. In particular, we study the limit behavior of $N_k$ - the number of critical points of $d_P$ with Morse index $k$ - as the density of points grows. We present explicit computations for the normalized, limiting, expectations and variances of the $N_k$, as well as distributional limit theorems. We link these results to recent results in which the Betti numbers of the random \\v{C}ech complex based on $P$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.4775","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"}