2010/10/11 by Steffen Winter, Winter, Steffen
Mathematics · #28A75 (Primary) #28A78 (Secondary) #28A80 #53C65 #FOS: Mathematics #Metric Geometry (math.MG) #math.MG #msc:28A75 #msc:28A78 #msc:28A80 #msc:53C65
paper · pdf · doi:10.48550/arxiv.1010.2032
17 pages, 3 figures
arxiv created 2010/10/11 · arxiv updated 2010/10/12
In some recent work, fractal curvatures Cfk(F) and fractal curvature measures Cfk(F, .), k = 0, ..., d, have been determined for all self-similar sets F in Rd, for which the parallel neighborhoods satisfy a certain regularity condition and a certain rather technical curvature bound. The regularity condition is conjectured to be always satisfied, while the curvature bound has recently been shown to fail in some concrete examples. As a step towards a better understanding of its meaning, we discuss several equivalent formulations of the curvature bound condition and also a very natural technically simpler condition which turns out to be stronger. These reformulations show that the validity this condition does not depend on the choice of the open set and the constant R appearing in the condition and allow to discuss some concrete examples of self-similar sets. In particular, it is shown that the class of sets satisfying the curvature bound condition is strictly larger than the class of sets satisfying the assumption of polyconvexity used in earlier results.