Alcoved polytopes. I.
The hyperplanes of the affine Coxeter arrangement subdivide \(\mathbb{R}^{n-1}\) into unit simplices, called alcoves. An alcoved polytope is a convex polytope which is the union of some alcoves. Hypersimplices, order polytopes and some special matroid polytopes are examples of alcoved polytopes. In Section 2 of the present paper the authors show the equivalence of the triangulation of hypersimplices due to \textit{R. P. Stanley} [Higher Comb., Proc. NATO Adv. Study Inst., Berlin (West) 1976, 49 (1977; Zbl 0359.05001)] and \textit{B. Sturmfels} [Gröbner bases and convex polytopes. University Lecture Series. 8. Providence, RI: American Mathematical Society (AMS). (1996; Zbl 0856.13020)], the alcove triangulation and the new ``circuit triangulation''. Than they extend this triangulations to general alcoved polytopes and give a formula for the volume of an alcoved polytope. Finally, they study three special examples of alcoved polytopes.
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