  In this paper we investigate the strict convexity and the differentiability
properties of the stable norm, which corresponds to the homogenized surface
tension for a periodic perimeter homogenization problem (in a regular and
uniformly elliptic case). We prove that it is always differentiable in totally
irrational directions, while in other directions, it is differentiable if and
only if the corresponding plane-like minimizers satisfying a strong Birkhoff
property foliate the torus. We also discuss the issue of the uniqueness of the
correctors for the corresponding homogenization problem.
