Anisotropic lower-dimensional Minkowski content and $\mathcal{S}$-content
Filip Fryš
TL;DR
The paper extends lower-dimensional Minkowski content and $\mathcal{S}$-content to anisotropic settings by introducing the $C$-anisotropic volume function $V_{E,C}(r)=\lambda^n(E\oplus rC)$ for a convex body $C$ and a compact set $E$. It proves that $V_{E,C}(\cdot)$ is a Kneser-type function of order $n$, enabling isotropic-type inequalities to hold in the anisotropic context and allowing the definition of $s$-dimensional $C$-anisotropic Minkowski content and $\mathcal{S}$-content, as well as corresponding Minkowski and $\mathcal{S}$-dimensions. The authors establish inequalities linking these anisotropic contents and show that the upper $\mathcal{S}$-dimension is independent of $C$, with $\dim_{\mathcal{M}}(E)^* = \dim^C_{\mathcal{S}}(E)^*$, while lower dimension relations may be strict. An application to the Sierpinski gasket with $D=\log_2(3)$ demonstrates nonexistence or strict inequality among anisotropic contents, highlighting the nuanced interplay between geometry and anisotropy. Together, these results provide a robust framework for anisotropic fractal analysis and dimension theory.
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 $\mathcal{S}$-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.