Quasi--Euclidean classification of Alcoved Convex Polyhedra

We give the quasi–Euclidean classification of the maximal (with respect to the $f$–vector) alcoved polyhedra. The $f$–vector of these maximal convex bodies is $(20,30,12)$, so they are simple dodecahedra. We find eight quasi–Euclidean classes. This classification, which preserves angles, is finer than the known combinatorial classification (found in 2012 by Jim'{e}nez and de la Puente), which has only six classes. Each alcoved polyhedron $\mathcal{P}$ is represented by a unique visualized idempotent matrix $A$. Some 2–minors of $A$ are invariants of $\mathcal{P}$: they are the tropical edge–lengths of $\mathcal{P}$.