Splitting Sandwiches Unevenly via Unique Sink Orientations and Rainbow Arrangements

TL;DR

This paper presents new combinatorial and topological proofs for the $ ext{α}$-Ham-Sandwich theorem, introduces rainbow arrangements as a generalization, and establishes the $ ext{∃} ext{R}$-completeness of their realizability problems.

Contribution

It provides novel proofs connecting the $ ext{α}$-Ham-Sandwich theorem with Unique Sink Orientations and introduces rainbow arrangements, extending the theorem beyond oriented matroids.

Findings

New combinatorial proof linking $ ext{α}$-Ham-Sandwich and Unique Sink Orientations

Generalization of the theorem to rainbow arrangements and oriented matroids

Proving $ ext{∃} ext{R}$-completeness of realizability problems for rainbow arrangements

Abstract

The famous Ham-Sandwich theorem states that any $d$ point sets in $\mathbb{R}^d$ can be simultaneously bisected by a single hyperplane. The $\alpha$-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 $\alpha$-Ham-Sandwich theorem and Unique Sink Orientations of grids. The second proof uses point-hyperplane duality and the Poincar\'e-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.