Equitable subdivisions within polygonal regions

We prove a generalization of the Ham-Sandwich Theorem. Specifically, let P be a simple polygonal region containing |R| = kn red points and |B| = km blue points in its interior with k ≥ 2. We show that P can be partitioned into k relatively-convex regions each of which contains exactly n red and m blue points. A region of P is relatively-convex if it is closed under geodesic (shortest) paths in P. We outline an O(kN2 log2 N) time algorithm for computing such a k-partition, where N = |R| + |B| + |P|.