2 research outputs found
Curvature-direction measures of self-similar sets
We obtain fractal Lipschitz-Killing curvature-direction measures for a large
class of self-similar sets F in R^d. Such measures jointly describe the
distribution of normal vectors and localize curvature by analogues of the
higher order mean curvatures of differentiable submanifolds. They decouple as
independent products of the unit Hausdorff measure on F and a self-similar
fibre measure on the sphere, which can be computed by an integral formula. The
corresponding local density approach uses an ergodic dynamical system formed by
extending the code space shift by a subgroup of the orthogonal group. We then
give a remarkably simple proof for the resulting measure version under minimal
assumptions.Comment: 17 pages, 2 figures. Update for author's name chang