Review of Steiner formulas in Fractal Geometry via Support measures and Complex Dimensions

Abstract: We review the theoretical framework that establishes a crucial bridge between the general Steiner-type formula of Hug, Last, and Weil and the theory of complex (fractal) dimensions of Lapidus et all. Two novel families of geometric functionals are introduced based on the support measures of the set itself as well as of its parallel sets, respectively. The associated scaling exponents provide new tools for extracting geometric information of the set, beyond its classical fractal dimensions while also encoding its outer Minkowski dimension. Furthermore the scaling exponents also directly connect to the complex dimensions of the set while preserving essential geometric information. The framework provides a fundamental link between measure-theoretic approaches and analytical methods in fractal geometry, offering new perspectives on both the geometric measure theory of singular sets and the complex analytic theory of fractal zeta functions.