Carathéodory’s Theorem in Depth

descriptionPublicationkeyboard_double_arrow_right Article , Preprint 11 May 2017Embargo end date: 01 Jan 2015 English Publisher:Springer Science and Business Media LLCJournal:Discrete & Computational Geometry, volume 58, pages 51-66 (issn: 0179-5376, eissn: 1432-0444,

Copyright policy )Funded by:EC | CONNECT

Authors: Fabila Monroy, Ruy; Huemer, Clemens;

doi: 10.1007/s00454-017-9893-8 , 10.48550/arxiv.1509.04575

arXiv: 1509.04575

handle: 2117/111689

Carathéodory’s Theorem in Depth

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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 $��_X(q)$ of points of $X$ in a halfspace containing $q$. In this paper we prove a depth version of Carath��odory's theorem. In particular, we prove that there exists a constant $c$ (that depends only on $d$ and $��_X(q)$) 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 depth versions of Helly's and Kirchberger's theorems.

Related Organizations

Universitat Politècnica de Catalunya
Spain
Universitat Polite`cnica de Catalunya
Spain
CINVESTAV
Mexico

Keywords

simplicial depth, Helly type theorem, Computational Geometry (cs.CG), FOS: Computer and information sciences, Simplicial depth, :32 Several complex variables and analytic spaces::32F Geometric convexity [Classificació AMS], Helly-type theorems and geometric transversal theory, Convex geometry, Àrees temàtiques de la UPC::Matemàtiques i estadística::Anàlisi matemàtica, :Matemàtiques i estadística::Anàlisi matemàtica [Àrees temàtiques de la UPC], FOS: Mathematics, Classificació AMS::32 Several complex variables and analytic spaces::32F Geometric convexity, Tukey depth, Computer Science - Computational Geometry, Mathematics - Combinatorics, Teoremes, Combinatorics (math.CO)

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