Curvature measures and fractals
Volume 453 / 2008
Abstract
\def\emph#1{{\it#1}}\def\eps{\varepsilon}\def\norm#1{\|#1\|}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 $\eps$-parallel sets \[ F_\eps := \{x\in\R^d : \inf_{y\in F} \norm{x-y}\le\eps\} \] instead, expecting that their limiting behaviour as $\eps\to 0$ provides information about the structure of the initial set $F$. In particular, we investigate the limiting behaviour of the total curvatures (or intrinsic volumes) $C_k(F_\eps), k=0,\ldots,d$, as well as weak limits of the corresponding curvature measures $C_k(F_\eps,\cdot)$ as $\eps\to 0$. This leads to the notions of \emph{fractal curvature} and \emph{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.
