A Remark on the Anisotropic Outer Minkowski content

TL;DR

This paper investigates an anisotropic variant of the outer Minkowski content for closed sets in Rn, establishing its existence and explicit form for sets where it matches the Hausdorff measure.

Contribution

It introduces an anisotropic outer Minkowski content, proves its existence on certain sets, and derives its explicit expression, extending classical geometric measure theory.

Findings

Existence of anisotropic outer Minkowski content for specific sets

Explicit formula for the anisotropic content

Extension of classical Minkowski content concepts

Abstract

We study an anisotropic version of the outer Minkowski content of a closed set in Rn. In particular, we show that it exists on the same class of sets for which the classical outer Minkowski content coincides with the Hausdorff measure, and we give its explicit form.