A new upper bound is given on the number of ways in which a set of N points in ℝn can be partitioned by k parallel hyperplanes. This bound improves upon a result of Olafsson and Abu-Mostafa (IEEE Trans. Pattern Anal. Machine Intell. 10 (2) (1988) 277); it agrees with the known (tight) result for the case k=1; and it is, for fixed k and n, tight to within a constant. A previously published claimed improvement to the bound of Olafsson and Abu-Mostafa is shown to be incorrect. © 2004 Elsevier B.V. All rights reserved.
Anthony, M. (2004). Partitioning points by parallel planes. Discrete Mathematics, 282(1–3), 17–21. https://doi.org/10.1016/j.disc.2003.10.025