Alcoved polytopes. I. (English) Zbl 1134.52019

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 R. P. Stanley [Higher Comb., Proc. NATO Adv. Study Inst., Berlin (West) 1976, 49 (1977; Zbl 0359.05001)] and 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.
Reviewer: Eike Hertel (Jena)


52B12 Special polytopes (linear programming, centrally symmetric, etc.)
52B11 \(n\)-dimensional polytopes
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