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Power savings for planar halving lines and k-sets

Improves the planar halving-line bound to

O(n^{4/3-\varepsilon})

for sets with no three collinear and an absolute ε > 0. More generally, an n-point set with no three collinear has

O(n(k+1)^{1/3-\varepsilon_0})

strictly separable k-subsets for

1\le k\le n/2

, with an absolute

\varepsilon_0\gt 0

. The constants and positive exponents are nonquantitative.

Manuscripts in this family · 1

Sep 25, 2026·main.pdf

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.

Full abstract