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 R d . 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.
A given order type in the plane can be represented by a point set. However, it might be difficult to recognize the orientations of some point triples. Recently, Aichholzer et al.[2] introduced exit graphs for visualizing order types in the plane. We present a new class of geometric graphs, called OT-graphs, using abstract order types and their axioms described in the well-known book by Knuth [14]. Each OTgraph corresponds to a unique order type. We develop efficient algorithms for recognizing OT-graphs and computing a minimal OT-graph for a given order type in the plane. We provide experimental results on all order types of up to nine points in the plane including a comparative analysis of exit graphs and OT-graphs.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.