Separating Bi-Chromatic Points by Parallel Lines

Given a 2-coloring of the vertices of a regular n-gon P , how many parallel lines are needed to separate the vertices into monochromatic subsets? We prove that bn/2c is a tight upper bound, and also provide anO(n log n) time algorithm to determine the direction that gives the minimum number of lines. If the polygon is a non-regular convex polygon, then n − 3 lines may be necessary, while n − 2 lines always suffice. This problem arises in machine learning and has implications about the representational capabilities of some neural networks.