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arXiv · 1606.08790

Robust Tverberg and colorful Carath\'eodory results via random choice

Pablo Soberón
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Abstract

We use the probabilistic method to obtain versions of the colorful Carath\'eodory 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 $R^d$, 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 the bound $N=rt+O(\sqrt{t})$ for fixed $r,d$ which is polynomial in each parameters. Our bounds extend to colorful versions of Tverberg's theorem, as well as Reay-type variations of this theorem.

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Pablo Soberón. 2016-06-28. Robust Tverberg and colorful Carath\'eodory results via random choice. https://arxiv.org/abs/1606.08790

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