Papers for

computational geometry developers

Papers whose findings have a practical use for this group, as judged from the abstract. Open a paper to read what it means in practice.

Mathematical results connect topological overlap and selection lemmas

Overlap-Helly theorems

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).

Wed 9 SeptComputational Geometry

The gist

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.

Open → 2609.10023v1