Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
self-similar set; parallel set; curvature measures; fractal curvatures; Minkowski content; Minkowski dimension; regularity condition; curvature bound condition
self-similar set; parallel set; curvature measures; fractal curvatures; Minkowski content; Minkowski dimension; regularity condition; curvature bound condition
Summary:
In some recent work, fractal curvatures $C^f_k(F)$ and fractal curvature measures $C^f_k(F,\cdot )$, $k= 0,\ldots ,d$, have been determined for all self-similar sets $F$ in $\mathbb R^d$, 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 of 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.
References:
[1] Cheeger J., Müller W., Schrader R.: On the curvature of piecewise flat spaces. Comm. Math. Phys. 92 (1984), 405–454. DOI 10.1007/BF01210729 | MR 0734226
[2] Bröcker L., Kuppe M.: Integral geometry of tame sets. Geom. Dedicata 82 (2000), 1897–1924. DOI 10.1023/A:1005248711077 | MR 1789065
[3] Falconer K.J.: On the Minkowski measurability of fractals. Proc. Am. Math. Soc. 123 (1995), no. 4, 1115–1124. DOI 10.1090/S0002-9939-1995-1224615-4 | MR 1224615 | Zbl 0838.28006
[4] Federer H.: Curvature measures. Trans. Amer. Math. Soc. 93 (1959), 418–491. DOI 10.1090/S0002-9947-1959-0110078-1 | MR 0110078 | Zbl 0089.38402
[5] Fu J.H.G.: Tubular neighborhoods in Euclidean spaces. Duke Math. J. 52 (1985), 1025–1046. DOI 10.1215/S0012-7094-85-05254-8 | MR 0816398 | Zbl 0592.52002
[6] Gatzouras D.: Lacunarity of self-similar and stochastically self-similar sets. Trans. Amer. Math. Soc. 352 (2000), no. 5, 1953–1983. DOI 10.1090/S0002-9947-99-02539-8 | MR 1694290 | Zbl 0946.28006
[7] Hutchinson J.E.: Fractals and self similarity. Indiana Univ. Math. J. 30 (1981), 713–747. DOI 10.1512/iumj.1981.30.30055 | MR 0625600 | Zbl 0598.28011
[8] Lapidus M.L., Pomerance C.: The Riemann zeta-function and the one-dimensional Weyl-Berry conjecture for fractal drums. Proc. London Math. Soc. 66 (1993), no. 1, 41–69. MR 1189091 | Zbl 0739.34065
[9] Llorente M., Winter S.: A notion of Euler characteristic for fractals. Math. Nachr. 280 (2007), no. 1–2, 152–170. DOI 10.1002/mana.200410471 | MR 2290389 | Zbl 1118.28006
[10] Rataj J., Winter S.: On volume and surface area of parallel sets. Indiana Univ. Math. J., to appear, online-preprint: http://www.iumj.indiana.edu/IUMJ/Preprints/4165.pdf
[11] Schief A.: Separation properties for self-similar sets. Proc. Amer. Math. Soc. 122 (1994), no. 1, 111–115. DOI 10.1090/S0002-9939-1994-1191872-1 | MR 1191872 | Zbl 0807.28005
[12] Winter S.: Curvature measures and fractals. Dissertationes Math. 453 (2008), 1–66. DOI 10.4064/dm453-0-1 | MR 2423952 | Zbl 1139.28300
[13] Winter S., Zähle M.: Fractal curvature measures of self-similar sets. submitted, arxiv.org/abs/1007.0696v2.
[14] Zähle M.: Curvatures and currents for unions of sets with positive reach. Geom. Dedicata 23 (1987), 155–171. DOI 10.1007/BF00181273 | MR 0892398
[15] Zähle M.: Approximation and characterization of generalized Lipschitz-Killing curvatures. Ann. Global Anal. Geom. 8 (1990), 249–260. DOI 10.1007/BF00127938 | MR 1089237
[16] Zähle M.: Lipschitz-Killing curvatures of self-similar random fractals. Trans. Amer. Math. Soc. 363 (2011), 2663–2684. DOI 10.1090/S0002-9947-2010-05198-0 | MR 2763731
Search
Partner of
