On the structure of the stable norm of periodic metrics

We study the differentiability of the stable norm ∥·∥ associated with a ℤⁿ periodic metric on ℝⁿ. Extending one of the main results of [Ba2], we prove that if p ∈ ℝⁿ and the coordinates of p are linearly independent over ℚ, then there is a linear 2-plane V containing p such that the restriction of ∥·∥ to V is differentiable at p. We construct examples where ∥·∥ it is not differentiable at a point with coordinates linearly independent over ℚ.