Robust Tverberg and Colourful Carathéodory Results via Random Choice

Abstract

We use the probabilistic method to obtain versions of the colorful 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 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 a bound N = rt + O( √ 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. MSC2010 Classification: 52A35, 05D40

The discrete yet ubiquitous theorems of Carathéodory, Helly, Sperner, Tucker, and Tverberg

“…Recently there have been two significant improvements, García-Colín et al [179] gave an asymptotically tight bound for the tolerant Tverberg theorem when the dimension and the size of the partition are fixed. Later, in [356], Soberón used the probabilistic method to give another asymptotic bound that is polynomial in all three parameters. Still we can ask for precise values.…”

A Sparse colorful polytopal KKM Theorem

“…A third application of Theorem 1.5 concerns a generalization of the colorful Carathéodory theorem [3]. The following sparse colorful version of the colorful Carathéodory theorem was proven in [7] and [15]. Theorem 1.11 (Holmsen [7], Soberón [15]).…”

“…The following sparse colorful version of the colorful Carathéodory theorem was proven in [7] and [15]. Theorem 1.11 (Holmsen [7], Soberón [15]). Let k be a positive integer and…”

New lower bounds for Tverberg partitions with tolerance in the plane

“…Several upper bounds for N (d, k, t) are known [8,10,16]. For example, Soberón [16] proved N (d, k, t) = kt + O( √ t) for fixed k and d. Therefore, it is interesting to find lower and upper bounds for P (d, k, t) = N (d, k, t) − kt. Theorems 5,6, and 7 provide new lower bounds for P (d, k, t) for d = 2.…”

“…Several upper bounds for N (d, k, t) are known [8,10,16]. For example, Soberón [16] Remark. Recently, we improved some lower bounds using computer pograms [5].…”