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.

翻译：著名的Ham-Sandwich定理指出，$\mathbb{R}^d$中的任意$d$个点集均可被单一超平面同时平分。$α$-Ham-Sandwich定理则为存在偏置切割（即截取各点集指定比例而非恰好一半的超平面）提供了充分条件。本文提出该定理的两种新证明。第一种证明完全基于组合方法，揭示了$α$-Ham-Sandwich定理与网格唯一汇定向之间的深刻联系。第二种证明运用点-超平面对偶性与Poincaré-Miranda定理，使结果可推广至定向拟阵乃至更一般结构。为此我们引入彩虹排列的新概念，该概念推广了着色伪超平面排列。研究过程中我们还证明：彩虹排列的可实现性判定属于$\exists \mathbb{R}$完全问题，这同时意味着网格唯一汇定向的可实现性判定也属于$\exists \mathbb{R}$完全问题。