Orthogonal equipartitions

Consider two absolutely continuous probability measures in the plane. A subdivision of the plane into k ≥ 2 regions is equitable if every region has weight 1/k in each measure. We show that, for any two probability measures in the plane and any integer k ≥ 2, there exists an equitable subdivision of the plane into k regions using at most k − 1 horizontal segments and at most k − 1 vertical segments. We also prove the existence of orthogonal equipartitions for point measures and present an efficient algorithm for computing an orthogonal equipartition.