Generalized Ham-Sandwich Cuts

Abstract

Bárány, Hubard, and Jerónimo recently showed that for given well-separated convex bodies S1,..., Sd in Rd 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 Rd which are 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≤{pipe}Pi{pipe}, finds the unique hyperplane h incident with a point in each Pi and having {pipe}h+∩Pi{pipe}=ai. Finally we show two other consequences of the direct combinatorial proof: the first is a stronger result, namely that in the discrete case, the conditions assuring existence and uniqueness of generalized cuts are also necessary; the second is an alternative and simpler proof of the theorem in Bárány et al., and in addition, we strengthen the result via a partial converse. © 2009 Springer Science+Business Media, LLC.