Existence of Anisotropic Minkowski Content

TL;DR

This paper investigates the conditions under which anisotropic Minkowski content exists for sets of finite perimeter, linking it to anisotropic perimeter properties of the set and its complement.

Contribution

It establishes a precise equivalence condition for the existence of anisotropic Minkowski content based on anisotropic perimeter relations.

Findings

Minkowski content of boundary equals perimeter under certain conditions

Existence of anisotropic outer Minkowski content implies existence of Minkowski content

Conditions relate anisotropic Minkowski content to anisotropic perimeter of set and complement

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.