Intersection patterns of convex sets via simplicial complexes, a survey

TL;DR

This survey reviews intersection patterns of convex sets using simplicial complexes, discussing their properties, computational complexity, and related Helly-type theorems, with a focus on recent advances and generalizations.

Contribution

It provides a comprehensive overview of intersection patterns via simplicial complexes, highlighting new results and the role of these notions in Helly-type theorems and good covers.

Findings

Differences among $d$-representable, $d$-collapsible, and $d$-Leray complexes clarified.

Computational complexity results for recognizing these complexes discussed.

New results on intersection patterns of good covers presented.

Abstract

The task of this survey is to present various results on intersection patterns of convex sets. One of main tools for studying intersection patterns is a point of view via simplicial complexes. We recall the definitions of so called $d$-representable, $d$-collapsible and $d$-Leray simplicial complexes which are very useful for this study. We study the differences among these notions and we also focus on computational complexity for recognizing them. A list of Helly-type theorems is presented in the survey and it is also discussed how (important) role play the above mentioned notions for the theorems. We also consider intersection patterns of good covers which generalize collections of convex sets (the sets may be `curvy'; however, their intersections cannot be too complicated). We mainly focus on new results.