No-dimensional Tverberg-type problems

TL;DR

This survey reviews recent advances in no-dimensional Tverberg problems, exploring convex hull intersections with small-radius balls, colorful variants, algorithmic challenges, and related matching problems in Euclidean space.

Contribution

It synthesizes recent progress, highlights open questions, and discusses algorithmic aspects of no-dimensional Tverberg-type problems and their variants.

Findings

Progress in solving no-dimensional Tverberg problems

Open questions on convex hull intersections

Connections to matching problems in Euclidean space

Abstract

Recently, Adiprasito et al. have initiated the study of the so-called no-dimensional Tverberg problem. This problem can be informally stated as follows: Given $n\geq k$, partition an $n$-point set in Euclidean space into $k$ parts such that their convex hulls intersect a ball of relatively small radius. In this survey, we aim to present the recent progress towards solving the no-dimensional Tverberg problem and new open questions arising in its context. Also, we discuss the colorful variation of this problem and its algorithmic aspects, particularly focusing on the case when each part of a partition contains exactly 2 points. The latter turns out to be related to the following no-dimensional Tverberg-type problem of Huemer et al.: For an even set of points in Euclidean space, find a perfect matching such that the balls with diameters induced by its edges intersect.