For a Riemannian polyhedra, we study the geometry of the unit ball for the unidimensional stable norm (stable ball). In the case of a unidimensional Riemannian polyhedra (graph), we show that the stable ball is a polytope whose vertices are completely described by combinatorial properties of the graph. We study then the realizable forms as stable ball of Riemannan manifolds of dimension larger than three. For a Riemannian manifold $(M, g)$ fixed, we show that a broad class of polytopes can appear as stable ball of metrics in the conformal class of $g$. We use for that a polyhedral technique.