Existence of Anisotropic Minkowski Content

Abstract: This paper is devoted to the existence of anisotropic Minkowski content and anisotropic outer Minkowski content. Our result is that the Minkowski content of the topological boundary of a given set of finite perimeter $E$ coincides with the perimeter of $E$ if and only if the anisotropic Minkowski content of the topological boundary of $E$ coincides with half of the sum of the anisotropic perimeter of $E$ and the anisotropic perimeter of the complement of $E$. As a consequence, we find that the existence of anisotropic outer Minkowski content of a given set of finite perimeter and its complement ensures the existence of outer Minkowski content of the set and its complement.