TITLE: Bi-partitioning $2n$ points on the plane with a straight line
QUESTION [9 upvotes]: Let $S$ be a set of $2n$ points in $\mathbb{R}^2$. Which is the maximum number of different bi-partitions of $S$ generated by a straight line?
More precisely, which is the maximum number of partitions $S=S_1 \sqcup S_2$ such that $|S_1|=n$ and there exists a line $\ell$ such that $S_1$ and $S_2$ and are the intersection of $S$ with the two open half-planes determined by $\ell$.

REPLY [11 votes]: These are called halving lines, and we don't know the exact order of their magnitude, just that it is between $\Omega(n2^{\sqrt\log n})$ and $O(n^{4/3})$.
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