Intersection patterns of convex sets via simplicial complexes: A survey

Abstract

The task of this survey is to present various results on intersection patterns of convex sets. One of the 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 also focus on computational complexity for recognizing them. A list of Helly-type theorems is presented in the survey. We also discuss the important role played by 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.