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We show that every bounded subset of an Euclidean space can be approximatedby a set that admits a certain vector field, the so-called Cahn-Hoffman vectorfield, that is subordinate to a given anisotropic metric and has asquare-integrable divergence. More generally, we introduce a concept of facetsas a kind of directed sets, and show that they can be approximated in a similarmanner.  We use this approximation to construct test functions necessary to prove thecomparison principle for viscosity solutions of the level set formulation ofthe crystalline mean curvature flow that were recently introduced by theauthors. As a consequence, we obtain the well-posedness of the viscositysolutions in an arbitrary dimension, which extends the validity of the resultin the previous paper.

Approximation of general facets by regular facets with respect to  anisotropic total variation energies and its application to the crystalline  mean curvature flow