Scaling exponents of curvature measures

Abstract

Fractal curvatures of a set $F\subset {\mathbb{R}}^d$ are roughly defined as suitably rescaled limits of the total curvatures of its parallel sets $F_{ϵ}$ as $ϵ$ tends to $0$ and have been studied in the last years in particular for self-similar and self-conformal sets. This previous work was focussed on establishing the existence of (averaged) fractal curvatures and related fractal curvature measures in the generic case when the curvature measures rescale like the volume of the parallel sets, that is, when the associated scaling exponents coincide with the Minkowski dimension of $F$. In the present paper we study the nongeneric situation when the scaling exponents do not coincide with the dimension. We demonstrate that the possibilities for nongeneric behaviour are rather limited and introduce the notion of local flatness, which allows a geometric characterization of nongenericy in $\mathbb{R}$ and ${\mathbb{R}}^2$. We expect local flatness to be characteristic also in higher dimensions. The results enlighten the geometric meaning of the scaling exponents.