Crossing patterns of semi-algebraic sets
Journal of Combinatorial Theory Series A
Opposite-Quadrant Depth in the Plane
Graphs and Combinatorics
Some Combinatorial and Algorithmic Applications of the Borsuk–Ulam Theorem
Graphs and Combinatorics
Points with large quadrant-depth
Proceedings of the twenty-sixth annual symposium on Computational geometry
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We show that for every @e0 there exists an angle @a=@a(@e) between 0 and @p, depending only on @e, with the following two properties: (1) For any continuous probability measure in the plane one can find two lines @?"1 and @?"2, crossing at an angle of (at least) @a, such that the measure of each of the two opposite quadrants of angle @p-@a, determined by @?"1 and @?"2, is at least 12-@e. (2) For any set P of n points in general position in the plane one can find two lines @?"1 and @?"2, crossing at an angle of (at least) @a and moreover at a point of P, such that in each of the two opposite quadrants of angle @p-@a, determined by @?"1 and @?"2, there are at least (12-@e)n-4 points of P.