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

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 $$\mathcal {S}$$ -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.