On a problem about quadrant-depth

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)=maxx∈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 . This answers a question raised by Stefan Felsner, and, as it turns out, also independently raised by Brönnimann, Lenchner, and Pach.