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 $Cn^{4/3-\varepsilon}$ 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.
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