  The real homology of a compact Riemannian manifold $M$ is naturally endowed
with the stable norm. The stable norm on $H_1(M,\mathbb{R})$ arises from the
Riemannian length functional by homogenization. It is difficult and interesting
to decide which norms on the finite-dimensional vector space
$H_1(M,\mathbb{R})$ are stable norms of a Riemannian metric on $M$. If the
dimension of $M$ is at least three, I. Babenko and F. Balacheff proved in
\cite{baba} that every polyhedral norm ball in $H_1(M,\mathbb{R})$, whose
vertices are rational with respect to the lattice of integer classes in
$H_1(M,\mathbb{R})$, is the stable norm ball of a Riemannian metric on $M$.
This metric can even be chosen to be conformally equivalent to any given
metric. The proof in \cite{baba} uses singular Riemannian metrics on polyhedra
which are finally smoothed. Here we present an alternative construction of such
metrics which remains in the geometric framework of smooth Riemannian metrics.
