Lipschitz-Killing curvatures of self-similar random fractals

Zähle, M.

doi:10.1090/S0002-9947-2010-05198-0

Lipschitz-Killing curvatures of self-similar random fractals
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Trans. Amer. Math. Soc. 363 (2011), 2663-2684 Request permission

Abstract:

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

References

J. D. Biggins, Martingale convergence in the branching random walk, J. Appl. Probability 14 (1977), no. 1, 25–37. MR 433619, DOI 10.2307/3213258
K. J. Falconer, Random fractals, Math. Proc. Cambridge Philos. Soc. 100 (1986), no. 3, 559–582. MR 857731, DOI 10.1017/S0305004100066299
Herbert Federer, Curvature measures, Trans. Amer. Math. Soc. 93 (1959), 418–491. MR 110078, DOI 10.1090/S0002-9947-1959-0110078-1
Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
William Feller, An introduction to probability theory and its applications. Vol. II, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR 0210154
Joseph Howland Guthrie Fu, Tubular neighborhoods in Euclidean spaces, Duke Math. J. 52 (1985), no. 4, 1025–1046. MR 816398, DOI 10.1215/S0012-7094-85-05254-8
Dimitris Gatzouras, Lacunarity of self-similar and stochastically self-similar sets, Trans. Amer. Math. Soc. 352 (2000), no. 5, 1953–1983. MR 1694290, DOI 10.1090/S0002-9947-99-02539-8
Siegfried Graf, Statistically self-similar fractals, Probab. Theory Related Fields 74 (1987), no. 3, 357–392. MR 873885, DOI 10.1007/BF00699096
Siegfried Graf, R. Daniel Mauldin, and S. C. Williams, The exact Hausdorff dimension in random recursive constructions, Mem. Amer. Math. Soc. 71 (1988), no. 381, x+121. MR 920961, DOI 10.1090/memo/0381
Daniel Hug, Günter Last, and Wolfgang Weil, A local Steiner-type formula for general closed sets and applications, Math. Z. 246 (2004), no. 1-2, 237–272. MR 2031455, DOI 10.1007/s00209-003-0597-9
John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747. MR 625600, DOI 10.1512/iumj.1981.30.30055
Peter Jagers, Branching processes with biological applications, Wiley Series in Probability and Mathematical Statistics—Applied Probability and Statistics, Wiley-Interscience [John Wiley & Sons], London-New York-Sydney, 1975. MR 0488341
Steven P. Lalley, The packing and covering functions of some self-similar fractals, Indiana Univ. Math. J. 37 (1988), no. 3, 699–710. MR 962930, DOI 10.1512/iumj.1988.37.37034
Marta Llorente and Steffen Winter, A notion of Euler characteristic for fractals, Math. Nachr. 280 (2007), no. 1-2, 152–170. MR 2290389, DOI 10.1002/mana.200410471
Russell Lyons, A simple path to Biggins’ martingale convergence for branching random walk, Classical and modern branching processes (Minneapolis, MN, 1994) IMA Vol. Math. Appl., vol. 84, Springer, New York, 1997, pp. 217–221. MR 1601749, DOI 10.1007/978-1-4612-1862-3_{1}7
Russell Lyons, Robin Pemantle, and Yuval Peres, Conceptual proofs of $L\log L$ criteria for mean behavior of branching processes, Ann. Probab. 23 (1995), no. 3, 1125–1138. MR 1349164
R. Daniel Mauldin and S. C. Williams, Random recursive constructions: asymptotic geometric and topological properties, Trans. Amer. Math. Soc. 295 (1986), no. 1, 325–346. MR 831202, DOI 10.1090/S0002-9947-1986-0831202-5
Olle Nerman, On the convergence of supercritical general (C-M-J) branching processes, Z. Wahrsch. Verw. Gebiete 57 (1981), no. 3, 365–395. MR 629532, DOI 10.1007/BF00534830
Norbert Patzschke, The strong open set condition in the random case, Proc. Amer. Math. Soc. 125 (1997), no. 7, 2119–2125. MR 1377002, DOI 10.1090/S0002-9939-97-03816-1
J. Rataj and S. Winter. On volume and surface area of parallel sets. Indiana Univ. Math. J. (to appear).
J. Rataj and M. Zähle, Normal cycles of Lipschitz manifolds by approximation with parallel sets, Differential Geom. Appl. 19 (2003), no. 1, 113–126. MR 1983898, DOI 10.1016/S0926-2245(03)00020-2
J. Rataj and M. Zähle, General normal cycles and Lipschitz manifolds of bounded curvature, Ann. Global Anal. Geom. 27 (2005), no. 2, 135–156. MR 2131910, DOI 10.1007/s10455-005-5218-x
Rolf Schneider, Curvature measures of convex bodies, Ann. Mat. Pura Appl. (4) 116 (1978), 101–134. MR 506976, DOI 10.1007/BF02413869
S. Winter. Curvature measures and fractals. Ph.D. thesis, University of Jena, 2006.
Steffen Winter, Curvature measures and fractals, Dissertationes Math. 453 (2008), 66. MR 2423952, DOI 10.4064/dm453-0-1
M. Zähle, Random processes of Hausdorff rectifiable closed sets, Math. Nachr. 108 (1982), 49–72. MR 695116, DOI 10.1002/mana.19821080105
M. Zähle, Curvature measures and random sets. II, Probab. Theory Relat. Fields 71 (1986), no. 1, 37–58. MR 814660, DOI 10.1007/BF00366271
M. Zähle, Integral and current representation of Federer’s curvature measures, Arch. Math. (Basel) 46 (1986), no. 6, 557–567. MR 849863, DOI 10.1007/BF01195026
M. Zähle, Approximation and characterization of generalised Lipschitz-Killing curvatures, Ann. Global Anal. Geom. 8 (1990), no. 3, 249–260. MR 1089237, DOI 10.1007/BF00127938