Mathematical results connect topological overlap and selection lemmas
Overlap-Helly theorems
Computational Geometry
Summary
The paper explores a new mathematical extension of a classical idea called Helly’s theorem, which helps figure out when multiple shapes or collections share a common point. The authors link this idea to other advanced theorems about overlapping shapes and selecting points, uniting some previous results in topology under a broader framework. This work also examines fractional versions of these results, shedding light on how many overlapping sets guarantee a shared point. Their findings offer new perspectives on continuous selection problems, which involve choosing points from complicated spaces in a consistent way.
What this means in practice
- •For topological data analysts: Use overlap theorems to identify shared features in complex datasets when calculating central points and intersections.
- •For computational geometry developers: Incorporate fractional Helly concepts to improve algorithms for finding common intersections in geometric configurations.
A theory result. No direct application yet.
Authors
Andreas F. Holmsen, Alfredo Hubard
Abstract
In this paper we introduce a generalization of Helly's theorem closely connected to Bárány-Gromov overlap theorems (also called selection lemmas). Our main result implies both the topological colorful Helly of Kalai and Meschulam and Karasev's topological centerpoint theorem. We further investigate the topological fractional Helly theorem from this overlap perspective, and show an overlap theorem for dense complexes (a continuous second selection lemma for tame maps).