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Isocanted alcoved polytopes. (English) Zbl 07285953
Summary: Through tropical normal idempotent matrices, we introduce isocanted alcoved polytopes, computing their $$f$$-vectors and checking the validity of the following five conjectures: Bárány, unimodality, $$3^d$$, flag and cubical lower bound (CLBC). Isocanted alcoved polytopes are centrally symmetric, almost simple cubical polytopes. They are zonotopes. We show that, for each dimension, there is a unique combinatorial type. In dimension $$d$$, an isocanted alcoved polytope has $$2^{d+1}-2$$ vertices, its face lattice is the lattice of proper subsets of $$[d+1]$$ and its diameter is $$d+1$$. They are realizations of $$d$$-elementary cubical polytopes. The $$f$$-vector of a $$d$$-dimensional isocanted alcoved polytope attains its maximum at the integer $$\lfloor d/3\rfloor$$.
##### MSC:
 52B12 Special polytopes (linear programming, centrally symmetric, etc.) 15A80 Max-plus and related algebras
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