Dense point sets with many halving lines

We construct a dense point set of n points in the plane with ne( √ logn) halving lines. This improves the bound O(n log n) of Edelsbrunner, Valtr and Welzl from 1997. We also observe that the upper bound on the maximum number of halving lines of dense point set can be improved to O(n). Our construction can be generalized to higher dimensions, for any d we construct a dense point set of n points in IR with nd−1e( √ logn) halving hyperplanes. Our lower bounds are asymptotically the same as the best known lower bounds for general point sets.