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arXiv · 2511.14526

A note on embracing exchange sequences in oriented matroids

Kristóf Bérczi·Benedek Nádor
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Abstract

An open problem in convex geometry asks whether two simplices $A,B\subseteq\mathbb{R}^d$, both containing the origin in their convex hulls, admit a polynomial-length sequence of vertex exchanges transforming $A$ into $B$ while maintaining the origin in the convex hull throughout. We propose a matroidal generalization of the problem to oriented matroids, concerning exchange sequences between bases under sign constraints on elements appearing in certain fundamental circuits. We formulate a conjecture on the minimum length of such a sequence, and prove it for oriented graphic matroids of directed graphs. We also study connections between our conjecture and several long-standing open problems on exchange sequences between pairs of bases in unoriented matroids.

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Kristóf Bérczi, Benedek Nádor. 2025-11-23. A note on embracing exchange sequences in oriented matroids. https://arxiv.org/abs/2511.14526

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