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Curvature-direction measures of self-similar sets

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Abstract

We obtain fractal Lipschitz–Killing curvature-direction measures for a large class of self-similar sets \(F\) in \(\mathbb{R }^{d}\). Such measures jointly describe the distribution of normal vectors and localize curvature by analogues of the higher order mean curvatures of differentiable sub-manifolds. 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.

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Authors and Affiliations

  1. Friedrich Schiller University Jena, Jena, Germany

    Tilman Johannes Bohl & Martina Zähle

Authors
  1. Tilman Johannes Bohl
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  2. Martina Zähle
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Correspondence to Tilman Johannes Bohl.

Additional information

Both supported by grant DFG ZA 242/5-1. The first author has previously worked under the name Tilman Johannes Rothe.

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Bohl, T.J., Zähle, M. Curvature-direction measures of self-similar sets. Geom Dedicata 167, 215–231 (2013). https://doi.org/10.1007/s10711-012-9810-5

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