{"paper":{"title":"Stars of Empty Simplices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Daniel Temesvari, Matthias Reitzner","submitted_at":"2018-08-27T08:39:45Z","abstract_excerpt":"Let $X=\\{x_1,\\ldots,x_n\\} \\subset \\mathbb R^d$ be an $n$-element point set in general position. For a $k$-element subset $\\{x_{i_1},\\ldots,x_{i_k}\\} \\subset X$ let the degree ${\\rm deg}_k(x_{i_1},\\ldots,x_{i_k})$ be the number of empty simplices $\\{x_{i_1},\\ldots,x_{i_{d+1}}\\} \\subset X$ containing no other point of $X$. The $k$-degree of the set $X$, denoted ${\\rm deg}_k(X)$, is defined as the maximum degree over all $k$-element subset of $X$.\n  We show that if $X$ is a random point set consisting of $n$ independently and uniformly chosen points from a compact set $K$ then ${\\rm deg}_d(X)=\\Th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.08734","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"}