Let φ:ℝ n → [0,+∞[ be a given positively one-homogeneous convex function, and let ?φ≔{φ≤ 1 }. Pursuing our interest in motion by crystalline mean curvature in three dimensions, we introduce and study the class ?φ (ℝ n ) of “smooth” boundaries in the relative geometry induced by the ambient Banach space (ℝ n , φ). It can be seen that, even when ?φ is a polytope, ?φ(ℝ n ) cannot be reduced to the class of polyhedral boundaries (locally resembling ∂?φ). Curved portions must be necessarily included and this fact (as well as the nonsmoothness of ∂?φ) is the source of several technical difficulties related to the geometry of Lipschitz manifolds. Given a boundary δE in the class ?φ(ℝ n ), we rigorously compute the first variation of the corresponding anisotropic perimeter, which leads to a variational problem on vector fields defined on δE. It turns out that the minimizers have a uniquely determined (intrinsic) tangential divergence on δE. We define such a divergence to be the φ-mean curvature κφ of δE; the function κφ is expected to be the initial velocity of δE, whenever δE is considered as the initial datum for the corresponding anisotropic mean curvature flow. We prove that κφ is bounded on δE and that its sublevel sets are characterized through a variational inequality.