Points with large α-depth

We show that for every ϵ > 0 there exists an angle α = α ( ϵ ) between 0 and π, depending only on ϵ, with the following two properties: (1) For any continuous probability measure in the plane one can find two lines ℓ 1 and ℓ 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 ℓ 1 and ℓ 2 , is at least 1 2 − ϵ . (2) For any set P of n points in general position in the plane one can find two lines ℓ 1 and ℓ 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 ℓ 1 and ℓ 2 , there are at least ( 1 2 − ϵ ) n − 4 points of P.