Equitable subdivisions within polygonal regions

Abstract

We prove a generalization of the Ham-Sandwich Theorem. Specifically, let P be a simple polygonal region containing | R | = k n red points and | B | = k m 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 ( k N 2 log 2 N ) time algorithm for computing such a k-partition, where N = | R | + | B | + | P | .