Equitable subdivisions within polygonal regions

Abstract

We prove a generalization of the Ham-Sandwich Theorem. Specifically, let P be a simple polygonal region containing $|R|=kn$ 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 $\mathrm{O}(k{N}^2{\log }^{2}N)$ time algorithm for computing such a k-partition, where $N=|R|+|B|+|P|$.