{"paper":{"title":"Caratheodory's Theorem in Depth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.CG","authors_text":"Clemens Huemer, Ruy Fabila-Monroy","submitted_at":"2015-09-15T14:19:51Z","abstract_excerpt":"Let $X$ be a finite set of points in $\\mathbb{R}^d$. The Tukey depth of a point $q$ with respect to $X$ is the minimum number $\\tau_X(q)$ of points of $X$ in a halfspace containing $q$. In this paper we prove a depth version of Carath\\'eodory's theorem. In particular, we prove that there exists a constant $c$ (that depends only on $d$ and $\\tau_X(q)$) and pairwise disjoint sets $X_1,\\dots, X_{d+1} \\subset X$ such that the following holds. Each $X_i$ has at least $c|X|$ points, and for every choice of points $x_i$ in $X_i$, $q$ is a convex combination of $x_1,\\dots, x_{d+1}$. We also prove dept"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.04575","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"}