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Robust Tverberg and Colourful Carathéodory Results via Random Choice

Part of: Extremal combinatorics General convexity

Published online by Cambridge University Press:  19 December 2017

PABLO SOBERÓN
PABLO SOBERÓN*
Affiliation:
Mathematics Department, Northeastern University, Boston, MA 02445, USA (e-mail: p.soberonbravo@northeastern.edu)
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Abstract

We use the probabilistic method to obtain versions of the colourful Carathéodory theorem and Tverberg's theorem with tolerance.

In particular, we give bounds for the smallest integer N = N(t,d,r) such that for any N points in ℝd, there is a partition of them into r parts for which the following condition holds: after removing any t points from the set, the convex hulls of what is left in each part intersect.

We prove a bound N = rt + O($\sqrt{t}$) for fixed r,d which is polynomial in each parameters. Our bounds extend to colourful versions of Tverberg's theorem, as well as Reay-type variations of this theorem.

MSC classification

Primary: 52A35: Helly-type theorems and geometric transversal theory
Secondary: 05D40: Probabilistic methods
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 27 , Issue 3 , May 2018 , pp. 427 - 440
Copyright
Copyright © Cambridge University Press 2017 

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