# Balanced $\epsilon$-separated partitioning by a hyperplane

Suppose we have $$m$$ points in $$R^n$$ and $$\epsilon>0$$ is a given constant. How can we find a hyperplane that the number of points that are $$\epsilon$$-close to it is minimum, with the constraint that it partitions the points to two sets of equal size. More formally, if the hyperplane is parametrized by $$u^\top x = b$$, in which $$||u||_2=1$$ is the unit vector orthogonal to the hyperplane. The constraint is that the number of points that is in the halfspace $$u^\top x is equal to the number of points in $$u^\top x. Among all the hyperplanes that satisfy this constraint, we want to minimize over points that are in the space $$|u^\top x - b|\le \epsilon$$. Is there a relatively fast algorithm (quadratic or cubic at most) algorithm that solves this efficiently?