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Discrete & Computational Geometry

, Volume 58, Issue 1, pp 51–66 | Cite as

Carathéodory’s Theorem in Depth

  • Ruy Fabila-Monroy
  • Clemens Huemer
Article

Abstract

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 Carathéodory’s theorem. In particular, we prove that there exist a constant c (that depends only on d and \(\tau _X(q)\)) and pairwise disjoint sets \(X_1,\ldots , X_{d+1} \subset X\) 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 \(x_1,\ldots , x_{d+1}\). We also prove depth versions of Helly’s and Kirchberger’s theorems.

Keywords

Helly type theorem Tukey depth Simplicial depth 

Mathematics Subject Classification

52A35 

Notes

Acknowledgements

We thank the anonymous referees whose comments helped us improve our paper significantly. R. Fabila-Monroy: Partially supported by Conacyt of Mexico Grant 253261. C. Huemery Partially supported by projects MTM2015-63791-R and Gen. Cat. DGR 2014SGR46. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 734922.

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Departament de MatemátiquesCinvestavMexicoMexico
  2. 2.Departament de MatemàtiquesUPCBarcelonaSpain

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