Martin Tancer
Published 2011-02-02, updated 2011-10-22Version 2
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.
Comments: 24 pages, 7 figures 2nd version: Lower bound for minimum k = k(d) such that every d-dimensional complex is k-representable was established in the meantime, therefore a reference is added. Several typos were fixed (including the definition of the Helly number)
Depth with respect to a family of convex sets
Piercing families of convex sets in the plane that avoid a certain subfamily with lines
Fractional Helly theorem for Cartesian products of convex sets
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