This is inspired by Mehul Chaturvedi's question Equally divided 10 points.
Consider all sets of 10 points in the plane, in which no 3 are collinear. A line \(l\) that connects 2 of these points is a dividing line if it divides the remaining points into 2 equal regions of 4 points each. Over all configurations, what is the maximum number of dividing lines?
By a simple counting argument, we can show that there are at least 5 dividing lines in any configuration.
If the 10 points form a convex set (ie no points are in the interior of the convex hull), then we can prove that there are exactly 5 dividing lines.
To get more than 5, consider the vertices of a 9-gon, along with the center. Then, any vertex that is connected to the center gives us a dividing line, and so there are (at least) 9 such lines.
Can we do better?
Can we generalize this to \( 10 = 2n \)?