  Through tropical normal idempotent matrices, we introduce isocanted alcoved
polytopes, computing their $f$--vectors and checking the validity of the
following five conjectures: B\'{a}r\'{a}ny, 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
$\lfloor d/3\rfloor$.
