Splitting Sandwiches Unevenly via Unique Sink Orientations and Rainbow Arrangements
Michaela Borzechowski, Sebastian Haslebacher, Hung P. Hoang, Patrick Schnider, Simon Weber

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 point sets in can be simultaneously bisected by a single hyperplane. The -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 -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…
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Advanced Graph Theory Research · Advanced Combinatorial Mathematics
