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May 29, 2026Open Access

Splitting Sandwiches Unevenly via Unique Sink Orientations and Rainbow Arrangements

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Authors

MBMichaela BorzechowskiSHSebastian HaslebacherHHHung P. Hoang

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Overview

New proofs demonstrate the α-Ham-Sandwich theorem in grid orientations and matroids, suggesting broader applications.

Key Points

  • This research aims to explore new proofs of the α-Ham-Sandwich theorem and its implications for unique sink orientations and rainbow arrangements.
  • Presented two distinct proofs for the α-Ham-Sandwich theorem.
  • Applied combinatorial techniques to connect the theorem with unique sink orientations.
  • Utilized point-hyperplane duality and the Poincaré-Miranda theorem to extend results to oriented matroids.
  • Established that the realizability problem for rainbow arrangements is ∃ℝ-complete.
  • Showed that the realizability problem for grid unique sink orientations is also ∃ℝ-complete.
  • Demonstrated a strong relationship between the α-Ham-Sandwich theorem and unique sink orientations.

Cite This Study

Borzechowski et al. (2026) studied this question.

synapsesocial.com/papers/6a192e68fab5b468c4417835https://doi.org/10.4230/lipics.socg.2026.19
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