Geometriae Dedicata

, Volume 167, Issue 1, pp 215–231 | Cite as

Curvature-direction measures of self-similar sets

  • Tilman Johannes Bohl
  • Martina Zähle
Original Paper


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.


Self-similar set Lipschitz–Killing curvature-direction measure Fractal curvature measure Minkowski content 

Mathematics Subject Classification (2000)

Primary: 28A80, 28A75, 37A99 Secondary: 28A78, 53C65 


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Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  1. 1.Friedrich Schiller University JenaJenaGermany

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