Handbook of discrete and computational geometry
Balanced partitions of two sets of points in the plane
Computational Geometry: Theory and Applications
2-Dimension Ham Sandwich Theorem for Partitioning into Three Convex Pieces
JCDCG '98 Revised Papers from the Japanese Conference on Discrete and Computational Geometry
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
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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|.