Quasi--Euclidean classification of Alcoved Convex Polyhedra

TL;DR

This paper classifies maximal alcoved convex polyhedra into eight quasi--Euclidean classes, refining previous combinatorial classifications, and introduces invariants based on idempotent matrices and tropical edge lengths.

Contribution

It provides a finer quasi--Euclidean classification of alcoved polyhedra, introducing matrix invariants and tropical lengths, extending prior combinatorial results.

Findings

Identified eight quasi--Euclidean classes of maximal alcoved polyhedra.

Established invariants using idempotent matrices and tropical edge lengths.

Refined the classification beyond previous combinatorial approaches.

Abstract

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 $\mathcal{P}$.