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Unconditional reflexive polytopes. (English) Zbl 07242486
A convex polytope \(P \subset \mathbb{R}^d\) is called unconditional if \((\sigma_1 p_1 , \dots, \sigma_d p_d) \in P\) for all \((p_1,\dots,p_d) \in P\) and all \((\sigma_1,\dots,\sigma_d)\in \{\pm 1\}^d\). The paper under review studies unconditional polytopes from the viewpoint of discrete geometry and combinatorial commutative algebra. In particular, it is shown that a lattice polytope \(P\) is an unconditional reflexive polytope if and only if \(P\) is obtained from the stable set polytope of a perfect graph. As examples, some special lattice polytopes are studied: type-B Birkhoff polytopes, signed Birkhoff polytopes, lattice polytopes arising from perfect CIS graphs, and unconditional chain polytopes of posets. Note that unconditional chain polytopes of posets were independently introduced in [H. Ohsugi and A. Tsuchiya, Isr. J. Math. 237, No. 1, 485–500 (2020; Zbl 07212851)]under the name “enriched chain polytopes”, and the same Gröbner bases of such polytopes were constructed there.
MSC:
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
52B12 Special polytopes (linear programming, centrally symmetric, etc.)
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