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.

翻译：凸几何中的一个开放问题提出：两个包含原点在其凸包内的单纯形 $A,B\subseteq\mathbb{R}^d$，是否存在一个多项式长度的顶点交换序列，将 $A$ 转换为 $B$，且在整个过程中保持原点位于凸包内。我们将该问题推广至有向拟阵，研究在特定基本回路中元素符号约束下基之间的交换序列。我们提出了关于此类序列最小长度的猜想，并对有向图的定向图拟阵证明了该猜想。我们还探讨了该猜想与无向拟阵中基对交换序列的若干长期开放问题之间的联系。