A power saving for planar halving lines

There are absolute constants ε > 0 and C such that every sufficiently large even n-point set in the plane with no three collinear has at most $C{n}^{4/3-\epsilon }$ unordered halving pairs. This gives a power saving over the classical $O(n^{4/3})$ bound for planar halving lines. The proof is nonquantitative and does not supply explicit constants.