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

TL;DR

This paper employs probabilistic methods to establish robust versions of the colorful Carathéodory and Tverberg theorems, providing bounds on the minimum number of points needed for intersection properties resilient to point removal.

Contribution

It introduces new bounds for Tverberg-type theorems with tolerance, extending to colorful and Reay-type variants using probabilistic techniques.

Findings

Bound N=rt+O(√t) for intersection after removing t points

Results extend to colorful Tverberg and Reay-type theorems

Bounds are polynomial in parameters for fixed r,d

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.