Geometry Seminar

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The alpha-Ham-Sandwich Theorem revisited

Speaker: Patrick Schnider, University of Basel and ETH Zürich

Location: Online

Date: Tuesday, April 7, 2026, 2 p.m.

Synopsis:

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.

This is joint work with Michaela Borzechowski, Sebastian Haslebacher, Hung Hoang, and Simon Weber.

Notes:

On Zoom.  Please contact Boris Aronov to be put on the email announcement list and obtain Zoom details.