Generalized Ham-Sandwich Cuts for Well Separated Point Sets

Bárány, Hubard, and Jerónimo recently showed that for given well separated convex bodies S1, . . . , Sd in R and constants βi ∈ [0, 1], there exists a unique hyperplane h with the property that Vol(h ∩Si) = βi·Vol(Si); h is the closed positive transversal halfspace of h, and h is a “generalized ham-sandwich cut”. We give a discrete analogue for a set S of n points in R which is partitioned into a family S = P1 ∪ · · · ∪ Pd of well separated sets and are in weak general position. The combinatorial proof inspires an O(n(log n)d−3) algorithm which, given positive integers ai ≤ |Pi|, finds the unique hyperplane h incident with a point in each Pi and having |h+ ∩ Pi| = ai. Finally we show that the conditions assuring existence and uniqueness of generalized cuts are also necessary.