Invariance of the normalized Minkowski content with respect to the ambient space

TL;DR

This paper demonstrates that the normalized Minkowski content of Minkowski measurable sets remains invariant across different ambient spaces, highlighting its intrinsic geometric property.

Contribution

It establishes the invariance of normalized Minkowski content for Minkowski measurable sets across various ambient spaces, extending known invariance properties of box dimensions.

Findings

Normalized Minkowski content is invariant under ambient space changes.

The invariance holds for Minkowski measurable sets.

The result emphasizes the intrinsic nature of Minkowski content.

Abstract

It is easy to show that the lower and the upper box dimensions of a bounded set in Euclidean space are invariant with respect to the ambient space. In this article we show that the Minkowski content of a Minkowski measurable set is also invariant with respect to the ambient space when normalized by an appropriate constant. In other words, the value of the normalized Minkowski content of a bounded, Minkowski measurable set is intrinsic to the set.