2-Dimension Ham Sandwich Theorem for Partitioning into Three Convex Pieces

Let m ≥ 2, n ≥ 2 and q ≥ 2 be positive integers. Let S r and S b be two disjoint sets of points in the plane such that no three points of S r ∪ S b are collinear, |S r | = nq, and |S b | = mq. This paper shows that Kaneko and Kano’s conjecture is true, i.e., S r ∪ S b can be partitioned into q subsets P 1,P 2,...,P q satisfying that: (i) conv(P i ) ∩ conv(P j ) = ∅ for all 1 ≤ i < j ≤ q; (ii) |P i ∩ S r |= n and |P i ∩ S b | = m for all 1 ≤ i ≤ q. This is a generalization of 2-dimension Ham Sandwich Theorem.