Scaling exponents of curvature measures

TL;DR

This paper investigates the scaling behavior of fractal curvatures in subsets of Euclidean space, especially in nongeneric cases where scaling exponents differ from the Minkowski dimension, introducing local flatness as a key concept.

Contribution

It characterizes nongeneric scaling behavior of fractal curvatures and introduces local flatness for geometric understanding in R and R^2.

Findings

Nongeneric behavior is limited and well-characterized.

Local flatness characterizes nongeneric cases in R and R^2.

Results provide geometric insights into scaling exponents.

Abstract

Fractal curvatures of a subset F of R^d are roughly defined as suitably rescaled limits of the total curvatures of its parallel sets F_e as e 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 k-th curvature measure C_k(F_e,.) scales like e^(k-D), where D ist 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 R and R^2. We expect local flatness to be characteristic also in higher dimensions. The results enlighten the geometric meaning of the scaling exponents.