Algorithms for Radon Partitions with Tolerance

Abstract

Let P be a set n points in a d-dimensional space. Tverberg theorem says that, if n is at least \((k-1)(d+1)\), 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. A tolerant Tverberg partition exists in any dimensions 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 \(\mathbb {R}^d\). Only few exact values of N(d, k, t) are known.