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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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