Support measures in fractal geometry

TL;DR

This paper introduces new geometric functionals called basic and support contents to analyze fractal properties of compact sets, linking support measures with fractal dimensions and providing deeper geometric insights.

Contribution

The paper develops two new families of geometric functionals derived from support measures, connecting them with fractal dimensions and curvatures, and introduces the concept of outer box dimension.

Findings

Support contents relate to fractal curvatures and complex dimensions.

Maximum basic scaling exponents connect to outer Minkowski dimension.

Support contents provide aggregated geometric information beyond classical measures.

Abstract

We introduce two novel families of geometric functionals-basic contents and support contents-for investigating the fractal properties of compact subsets in Euclidean space. These functionals are derived from the support measures arising in connection with the general Steiner formula due to Hug, Last, and Weil, and offer new tools for extracting geometric information beyond classical fractal dimensions. The basic contents are constructed from the support measures of the set itself, while the support contents arise from those of its parallel sets. Associated scaling exponents characterize the asymptotic behavior of these measures as the resolution parameter tends to zero. We establish a fundamental connection between the maximum of the basic scaling exponents and the outer Minkowski dimension. The proof relies on the novel notion of outer box dimension. Furthermore, we explore how support contents supply aggregated information on geometric features that is provided separately by the basic contents, and how they relate to fractal curvatures and complex dimensions. Our results provide a deeper understanding of which geometric aspects actually contribute to fractal invariants such as the (outer) Minkowski content. We provide some illustrative examples.