Opposite-Quadrant Depth in the Plane
Graphs and Combinatorics
Points with large quadrant-depth
Proceedings of the twenty-sixth annual symposium on Computational geometry
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Let P be a set of n points in general position in the plane. For every x@?P let D(x,P) be the maximum number such that there are at least D(x,P) points of P in each of two opposite quadrants determined by some two perpendicular lines through x. Define D(P)=max"x"@?"PD(x,P). In this paper we show that D(P)=c|P| for every set P in general position in the plane where c is an absolute constant that is strictly greater than 18. This answers a question raised by Stefan Felsner, and, as it turns out, also independently raised by Bronnimann, Lenchner, and Pach.