I have the following question regarding a generalization of the classical theorem of 'Ham-Sandwich Cut' for pseudo point-configuration.
First, a few definitions. A collection of continuous curves in the plane is called a pseudo-line arrangement if each curve is $x$-monotone and every pair of curves intersect at most one point. Now, a pseudo configuration of points is defined by a pair $(P,S)$, where $P$ is a set points in $\mathbb{R}^2$ and $S$ is a pseudo-line arrangement, such that, for every pair of points in $P$ there is a unique curve in $S$ passing through both of them. Note that $|S|={|P|\choose 2}$.
Coming to the main question, let $(P,S)$ be a pseudo-configuration of points in the plane such that $|P|=4n$ and $P$ is the union of two $2n$-element sets, $P_1$ and $P_2$ separated by a straight line $\ell$. Then, can we find a curve $c$, in $O(n)$ time, such that, $S\cup \{\ell, c\}$ is a pseudo-line arrangement and on both sides of $c$ there are equal number ($n$) of points of $P_i$ ($i=1,2$)?
