Abstract
Fractal curvatures of a compact set F in Rd are roughly defined as suitably rescaled limits of the total curvatures of its parallel sets of F as diametr tends to 0 and have been studied in the last years in particular for self-similar and self-conformal sets. In the present paper we study the nongeneric situation when the scaling exponents are not determined by the dimension of F.
We demonstrate that the possibilities for nongeneric behaviour are rather limited and introduce the notion of local flatness, which allows a geometric characterization of nongenericity in R and R2. We expect local flatness to be characteristic also in higher dimensions.
The results enlighten the geometric meaning of the scaling exponents.