A characterization of convex calibrable sets in R^N with respect to anisotropic norms

A set is called "calibrable" if its characteristic function is an eigenvector of the subgradient of the total variation. The main purpose of this paper is to characterize the "phi-calibrability" of bounded convex sets in R(N) with respect to a norm phi (called anisotropy in the sequel) by the anisotropic mean phi-curvature of its boundary. It extends to the anisotropic and crystalline cases the known analogous results in the Euclidean case. As a by-product of our analysis we prove that any convex body C satisfying a phi-ball condition contains a convex phi-calibrable set K such that, for any V is an element of [vertical bar K vertical bar,vertical bar C vertical bar], the subset of C of volume V which minimizes the phi-perimeter is unique and convex. We also describe the anisotropic total variation flow with initial data the characteristic function of a bounded convex set.