Before revision:  We also observe that the upper bound on the maximum number of halving lines of dense point set can be improved to O(n^{7/6 Our construction can be generalized to higher dimensions, for any d we construct a dense point set of n points in R^d with n^{d-1}e^{\Omega\left({\log n}\right)} halving hyperplanes.
After revision:  Our construction can be generalized to higher dimensions, for any d we construct a dense point set of n points in R^d with n^{d-1}e^{\Omega\left({\log n}\right)} halving hyperplanes.