3 research outputs found
Tverberg partitions as weak epsilon-nets
We prove a Tverberg-type theorem using the probabilistic method. Given
, we find the smallest number of partitions of a set in
into parts needed in order to induce at least one Tverberg partition
on every subset of with at least elements. This
generalizes known results about Tverberg's theorem with tolerance.Comment: 14 page
Tolerance for colorful Tverberg partitions
Tverberg's theorem bounds the number of points needed for the
existence of a partition into parts whose convex hulls intersect. If the
points are colored with colors, we seek partitions where each part has at
most one point of each color. In this manuscript, we bound the number of color
classes needed for the existence of partitions where the convex hulls of the
parts intersect even after any set of colors is removed. We prove
asymptotically optimal bounds for when , improve known bounds
when , and give a geometric characterization for the configurations of
points for which .Comment: 13 pages, 1 figur
New Lower Bounds for Tverberg Partitions with Tolerance in the Plane
Let be a set points in a -dimensional space. Tverberg's theorem
says that, if is at least , then can be partitioned into
sets whose convex hulls intersect. Partitions with this property are called
{\em Tverberg partitions}. A partition has tolerance if the partition
remains a Tverberg partition after removal of any set of points from .
Tolerant Tverberg partitions exist in any dimension provided that is
sufficiently large. Let be the smallest value of such that
tolerant Tverberg partitions exist for any set of points in .
Only few exact values of are known.
In this paper we establish a new tight bound for . We also prove
many new lower bounds on for and .Comment: 10 figure