Carathodory's theorem in depth
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European Commission's projectCONNECT - Combinatorics of Networks and Computation (EC-H2020-734922)
Let X be a finite set of points in RdRd . The Tukey depth of a point q with respect to X is the minimum number tX(q)tX(q) of points of X in a halfspace containing q. In this paper we prove a depth version of Carathéodory’s theorem. In particular, we prove that there exist a constant c (that depends only on d and tX(q)tX(q) ) and pairwise disjoint sets X1,…,Xd+1¿XX1,…,Xd+1¿X such that the following holds. Each XiXi has at least c|X| points, and for every choice of points xixi in XiXi , q is a convex combination of x1,…,xd+1x1,…,xd+1 . We also prove depth versions of Helly’s and Kirchberger’s theorems.
CitationFabila, R., Huemer, C. Carathodory's theorem in depth. "Discrete and computational geometry", 1 Juliol 2017, vol. 58, núm. 1, p. 51-66.