Anisotropic lower-dimensional Minkowski content and $\mathcal{S}$-content

Abstract

This paper investigates the lower-dimensional anisotropic Minkowski content and $\mathcal{S}$-content. We establish that these anisotropic contents exhibit properties analogous to their isotropic counterparts by proving analogous inequalities between the lower-dimensional anisotropic Minkowski content and $\mathcal{S}$-content. A key component of our approach is demonstrating that the associated anisotropic volume function is of Kneser type, a result that underpins many of our proofs. In addition, we introduce anisotropic versions of the Minkowski and $\mathcal{S}$-dimensions and derive inequalities relating them. As an application, we analyze the existence of the $\log_2(3)$-dimensional anisotropic Minkowski and $\mathcal{S}$-contents of the Sierpinski gasket.