Points with large α-depth

摘要

We show that for every ε{lunate} > 0 there exists an angle α = α (ε{lunate}) between 0 and π, depending only on ε{lunate}, 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) α, such that the measure of each of the two opposite quadrants of angle π - α, determined by ?_1 and ?_2, is at least frac(1, 2) - ε{lunate}. (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) α and moreover at a point of P, such that in each of the two opposite quadrants of angle π - α, determined by ?_1 and ?_2, there are at least (frac(1, 2) - ε{lunate}) n - 4 points of P.