Lipschitz-Killing curvatures of self-similar random fractals

2010/09/30 by Martina Zähle, Zähle, Martina · 4 citations
Mathematics · #28A78 #53C65 #60D05 #60J80 #60J85 #FOS: Mathematics #Metric Geometry (math.MG) #Primary 28A80 #Probability (math.PR) #Secondary 28A75 #math.MG #math.PR #msc:28A75 #msc:28A78 #msc:28A80 #msc:53C65 #msc:60D05 #msc:60J80 #msc:60J85

paper · pdf · doi:10.48550/arxiv.1009.6166

22 page

arxiv created 2010/09/30 · arxiv updated 2010/10/01

Abstract

For a large class of self-similar random sets F in R^d geometric parameters C_k(F), k=0,...,d, are introduced. They arise as a.s. (average or essential) limits of the volume C_d(F(ε)), the surface area C_d-1(F(ε)) and the integrals of general mean curvatures over the unit normal bundles C_k(F(ε)) of the parallel sets F(ε) of distance ε, rescaled by ε^D-k, as ε→ 0. Here D equals the a.s. Hausdorff dimension of F. The corresponding results for the expectations are also proved.

Cited by

• Review of Steiner formulas in Fractal Geometry via Support measures and Complex Dimensions
• Support measures in fractal geometry
• Recent Progress on Fractal Percolation
• Mean Minkowski content and mean fractal curvatures of random self-similar code tree fractals

Related

• Mean Lipschitz-Killing curvatures for homogeneous random fractals
• Fractal curvature measures of self-similar sets
• Mean Minkowski content and mean fractal curvatures of random self-similar code tree fractals
• Curvature densities of self-similar sets
• Fractal curvatures and Minkowski content of self-conformal sets