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Hyperplane arrangements, discrete geometry, convex polytopes, and optimization. For more general questions concerning matroids, use the matroid-theory tag.

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What are / could be the applications of Delaunay oriented matroids?

The Delaunay oriented matroid is studied in detail by F. Santos (actually this paper is more general). For a set of points $S$ (in any dimension), let $C$ be a sphere, $C^+$ be its interior, $C^-$ be …