Points with large α-depth

We show that for any ǫ> 0 there exists an angle α = α(ǫ) between 0 and π, depending only on ǫ, with the following two properties: (1) For any continuous probability measure in the plane one can find two lines ℓ1 and ℓ2, 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 ℓ1 and ℓ2, is at least 1 2 − ǫ. (2) For any set P of n points in general position in the plane one can find two lines ℓ1 and ℓ2, 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 ℓ1 and ℓ2, there are at least