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Carathodory's theorem in depth

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1509.04575 (1).pdf (455,0Kb)
 
10.1007/s00454-017-9893-8
 
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hdl:2117/111689

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Fabila Monroy, Ruy
Huemer, ClemensMés informacióMés informacióMés informació
Document typeArticle
Defense date2017-07-01
Rights accessOpen Access
Attribution-NonCommercial-NoDerivs 3.0 Spain
This work is protected by the corresponding intellectual and industrial property rights. Except where otherwise noted, its contents are licensed under a Creative Commons license : Attribution-NonCommercial-NoDerivs 3.0 Spain
ProjectCONNECT - Combinatorics of Networks and Computation (EC-H2020-734922)
Abstract
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. 
URIhttp://hdl.handle.net/2117/111689
DOI10.1007/s00454-017-9893-8
ISSN0179-5376
Publisher versionhttps://link.springer.com/article/10.1007%2Fs00454-017-9893-8
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  • DCCG - Grup de recerca en geometria computacional, combinatoria i discreta - Articles de revista [85]
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