Discrete Helly-type theorems for pseudohalfplanes

03/20/2021

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by   Balázs Keszegh, et al.

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We prove discrete Helly-type theorems for pseudohalfplanes, which extend recent results of Jensen, Joshi and Ray about halfplanes. Among others we show that given a family of pseudohalfplanes H and a set of points P, if every triple of pseudohalfplanes has a common point in P then there exists a set of at most two points that hits every pseudohalfplane of H. We also prove that if every triple of points of P is contained in a pseudohalfplane of H then there are two pseudohalfplanes of H that cover all points of P. To prove our results we regard pseudohalfplane hypergraphs, define their extremal vertices and show that these behave in many ways as points on the boundary of the convex hull of a set of points. Our methods are purely combinatorial.