New lower bounds for Tverberg partitions with tolerance in the plane

Abstract

Let P be a set n points in a d-dimensional space. Tverberg’s theorem says that, if n is at least (k−1)(d+1)+1, then P can be partitioned into k sets whose convex hulls intersect. Partitions with this property are called Tverberg partitions. A partition has tolerance t if the partition remains a Tverberg partition after removal of any set of t points from P. Tolerant Tverberg partitions exist in any dimension provided that n is sufficiently large. Let N(d,k,t) be the smallest value of n such that tolerant Tverberg partitions exist for any set of n points in Rd. Only few exact values of N(d,k,t) are known.In this paper we establish a new tight bound for N(2,2,2). We also prove many new lower bounds on N(2,k,t) for k≥2 and t≥1.