Intersection Patterns of Convex Sets via Simplicial Complexes: A Survey

Martin Tancer ()
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Martin Tancer: Charles University, Faculty of Mathematics and Physics, Department of Applied Mathematics and Institute for Theoretical Computer Science

A chapter in Thirty Essays on Geometric Graph Theory, 2013, pp 521-540 from Springer

Abstract: 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. Mathematics Subject Classification (2010): primary 52A35, secondary 05E45, 52A20

Keywords: Simplicial Complex; Good Cover; Intersection Pattern; Color Class; Nonempty Intersection (search for similar items in EconPapers)
Date: 2013
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4614-0110-0_28

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DOI: 10.1007/978-1-4614-0110-0_28

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