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Advances in Geometry

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Fractal curvature measures of self-similar sets

Steffen Winter
  • Karlsruhe Institute of Technology, Department of Mathematics, 76133 Karlsruhe, Germany
  • :
/ Martina Zähle
  • University of Jena, Mathematical Institute, 07737 Jena, Germany
  • :
Published Online: 2013-03-29 | DOI: https://doi.org/10.1515/advgeom-2012-0026

Abstract

Fractal Lipschitz-Killing curvature measures Ckf (F; ·), k = 0; ...; d, are determined for a large class of self-similar sets F in ℝd. They arise as weak limits of the appropriately rescaled classical Lipschitz-Killing curvature measures Ck(Fε; ·) from geometric measure theory of parallel sets Fε for small distances ε. Due to self-similarity the limit measures appear to be constant multiples of the normalized Hausdorff measures on F, and the constants agree with the corresponding total fractal curvatures Ckf (F) This provides information on the ’second order’ geometric fine structure of such fractals.

Keywords: Self-similar set; parallel set; curvature measures; Lipschitz-Killing curvature measures; Minkowski content; Minkowski dimension; fractal curvature measures

The first author was supported by grant DFG WE 1613/2-1 and the second author by grant DFG ZA 242/5-1.


Published Online: 2013-03-29

Published in Print: 2013-04-01


Citation Information: Advances in Geometry. Volume 13, Issue 2, Pages 229–244, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2012-0026, March 2013

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[1]
Evgeny Spodarev, Peter Straka, and Steffen Winter
Chaos, Solitons & Fractals, 2015, Volume 75, Page 134

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