Curvature measures and fractals.

*(English)*Zbl 1139.28300Summary: Curvature measures are an important tool in geometric measure theory and other fields of mathematics for describing the geometry of sets in Euclidean space. But the ‘classical’ concepts of curvature are not directly applicable to fractal sets. We try to bridge this gap between geometric measure theory and fractal geometry by introducing a notion of curvature for fractals. For compact sets \(F\subseteq {\mathbb R}^d\) (e.g. fractals), for which classical geometric characteristics such as curvatures or Euler characteristic are not available, we study these notions for their \(\varepsilon\)-parallel sets
\[
F_\varepsilon := \{x\in{\mathbb R}^d : \inf_{y\in F} \|x-y\|\leq\varepsilon\}
\]
instead, expecting that their limiting behaviour as \(\varepsilon\to 0\) provides information about the structure of the initial set \(F\). In particular, we investigate the limiting behaviour of the total curvatures (or intrinsic volumes) \(C_k(F_\varepsilon), k=0,\ldots,d\), as well as weak limits of the corresponding curvature measures \(C_k(F_\varepsilon,\cdot)\) as \(\varepsilon\to 0\). This leads to the notions of fractal curvature and fractal curvature measure, respectively. The well known Minkowski content appears in this context as one of the fractal curvatures. For certain classes of self-similar sets, results on the existence of (averaged) fractal curvatures are presented. These limits can be calculated explicitly and are in a certain sense ‘invariants’ of the sets, which may help to distinguish and classify fractals. Based on these results also the fractal curvature measures of these sets are characterized. As a special case and a significant refinement of known results, a local characterization of the Minkowski content is given.