Generalized Ham-Sandwich Cuts

Bárány, Hubard, and Jerónimo recently showed that for given well-separated convex bodies S 1 ,…, S d in R d and constants β i ∈[0,1], there exists a unique hyperplane h with the property that Vol ( h + ∩ S i )= β i ⋅Vol ( S i ); 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 d which are partitioned into a family S = P 1 ∪ ⋅⋅⋅ ∪ P d 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 a i ≤| P i |, finds the unique hyperplane h incident with a point in each P i and having | h + ∩ P i |= a i . 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.