Anisotropic Minkowski Content for Countably $\mathcal{H}^k$-rectifiable Sets

TL;DR

This paper proves the existence of anisotropic Minkowski content for certain rectifiable sets, showing it depends on the set's rectifiability and the choice of anisotropic norm.

Contribution

It establishes the existence and characterization of anisotropic Minkowski content for countably rectifiable sets under specific conditions.

Findings

Existence of $C$-anisotropic Minkowski content for $k$-rectifiable sets.

Coincidence of Minkowski content with a natural functional depending on $C$.

Dependence of content existence on the choice of $C$.

Abstract

This paper investigates the existence of the anisotropic lower-dimensional Minkowski content. We establish that the $C$-anisotropic $k$-dimensional Minkowski content of a $k$-rectifiable compact set always exists and coincides with a specific functional that depends naturally on $C$. We further show that the same conclusion holds for countably $\mathcal{H}^k$-rectifiable compact sets, provided that the so-called \emph{AFP-condition} is satisfied. In addition, we discuss how the existence of the $C$-anisotropic $k$-dimensional Minkowski content for a countably $\mathcal{H}^k$-rectifiable compact set depends on the choice of $C$.