Carathodory's theorem in depth

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...

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Bibliographic Details
Main Authors:	Fabila Monroy, Ruy, Huemer, Clemens
Format:	Publication
Language:	English
Published:	01.07.2017
Subjects:	32 Several complex variables and analytic spaces 32F Geometric convexity Anàlisi matemàtica Classificació AMS Convex geometry Helly type theorem Matemàtiques i estadística Simplicial depth Teoremes Tukey depth Àrees temàtiques de la UPC
ISSN:	0179-5376
Online Access:	Get full text
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Summary:	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.
ISSN:	0179-5376
DOI:	10.1007/s00454-017-9893-8