No-dimensional Tverberg-type problems

2025-06-16

AI Summary

This paper investigates the dimension-free Tverberg problem: partitioning an $n$-point set in Euclidean space into $k$ subsets such that the intersection of their convex hulls contains a ball of minimal possible radius. It provides the first systematic survey of the problem’s development and introduces a novel “matching-type” variant—where $k = n/2$ and each subset consists of exactly two points—requiring that the diameter balls induced by all matching edges share a common point. This variant is shown to be equivalent to the colorful Tverberg-type matching problem proposed by Huemer et al. Employing techniques from combinatorial geometry, convex analysis, probabilistic methods, discrete topology, and algorithm design, the authors derive tight upper bounds on the optimal radius, establish necessary and sufficient conditions for intersection nonemptiness, and devise a polynomial-time approximation algorithm. These results deepen the understanding of interplay between geometric structure and combinatorial constraints, advancing the interface of high-dimensional discrete and computational geometry.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

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 $ngeq k$, 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.

Problem

• Partition n points into k intersecting convex hulls
• Study colorful variations and algorithmic aspects
• Find perfect matching with intersecting diameter balls

Innovation

• Partition point sets into intersecting convex hulls
• Study colorful variations and algorithmic aspects
• Focus on perfect matchings with intersecting balls