Splitting Sandwiches Unevenly via Unique Sink Orientations and Rainbow Arrangements

The famous Ham-Sandwich theorem states that any $d$ point sets in $\mathbb{R}^d$ can be simultaneously bisected by a single hyperplane. The $α$-Ham-Sandwich theorem gives a sufficient condition for the existence of biased cuts, i.e., hyperplanes that do not cut off half but some prescribed fraction of each point set. We give two new proofs for this theorem. The first proof is completely combinatorial and highlights a strong connection between the $α$-Ham-Sandwich theorem and Unique Sink Orientations of grids. The second proof uses point-hyperplane duality and the Poincaré-Miranda theorem and allows us to generalize the result to and beyond oriented matroids. For this we introduce a new concept of rainbow arrangements, generalizing colored pseudo-hyperplane arrangements. Along the way, we also show that the realizability problem for rainbow arrangements is $\exists \mathbb{R}$-complete, which also implies that the realizability problem for grid Unique Sink Orientations is $\exists \mathbb{R}$-complete.

💡 Research Summary

The paper revisits the α‑Ham‑Sandwich theorem, which generalizes the classic Ham‑Sandwich theorem by allowing each of d point sets (or mass distributions) in ℝⁿ to be cut at a prescribed fraction αᵢ rather than exactly in half. The authors present two completely new proofs and, in the process, uncover deep connections to combinatorial geometry and computational complexity.

First proof – Grid Unique‑Sink Orientations (USO).
Given well‑separated point families P₁,…,P_d in weak general position, they construct a d‑dimensional grid Γ_P whose dimensions are the cardinalities |P_i|. Vertices correspond to tuples (a₁,…,a_d) selecting one point from each set. For each edge that changes only the i‑th coordinate, they compare the two candidate points p_{i,a_i} and p_{i,a’_i} via the “colorful hyperplane” spanned by the other selected points. The direction of the edge is set according to which of the two points lies above the other hyperplane. This yields an orientation σ_P of the grid.

Using the continuous version of the α‑Ham‑Sandwich theorem for convex bodies (Bárány, Hubard, Jerónimo), they show that every induced subgrid possesses a sink, namely a (1,…,1)‑cut. Moreover, any sub‑cube (a subgrid where each dimension contains at most two vertices) contains all possible out‑maps, which guarantees that each sub‑cube is a USO. Lemma 5 then lifts this property to all subgrids, proving that σ_P is a grid USO.

A classic result on USOs (