Concentration of 1-Lipschitz maps into an infinite dimensional ℓ^{𝑝}-ball with the ℓ^{𝑞}-distance function

Funano, Kei

doi:10.1090/S0002-9939-09-09873-6

Concentration of $1$-Lipschitz maps into an infinite dimensional $\ell ^p$-ball with the $\ell ^q$-distance function
HTML articles powered by AMS MathViewer

by Kei Funano

Proc. Amer. Math. Soc. 137 (2009), 2407-2417

DOI: https://doi.org/10.1090/S0002-9939-09-09873-6

Published electronically: March 12, 2009

PDF | Request permission

Abstract:

In this paper, we study the Lévy-Milman concentration phenomenon of $1$-Lipschitz maps into infinite dimensional metric spaces. Our main theorem asserts that the concentration to an infinite dimensional $\ell ^p$-ball with the $\ell ^q$-distance function for $1\leq p<q\leq +\infty$ is equivalent to the concentration to the real line.

References

K. Funano, Asymptotics of mm-spaces, doctoral thesis, in preparation.
K. Funano, Central and $L^p$-concentration of $1$-Lipschitz maps into $\mathbb {R}$-trees, to appear in J. Math. Soc. Japan.
K. Funano, Concentration of maps and group action, preprint, available online at http://front.math.ucdavis.edu/0807.3210, 2008.
Kei Funano, Observable concentration of mm-spaces into spaces with doubling measures, Geom. Dedicata 127 (2007), 49–56. MR 2338515, DOI 10.1007/s10711-007-9156-6
K. Funano, Observable concentration of mm-spaces into nonpositively curved manifolds, preprint, available online at http://front.math.ucdavis.edu/0701.5535, 2007.
Thierry Giordano and Vladimir Pestov, Some extremely amenable groups, C. R. Math. Acad. Sci. Paris 334 (2002), no. 4, 273–278 (English, with English and French summaries). MR 1891002, DOI 10.1016/S1631-073X(02)02218-5
Thierry Giordano and Vladimir Pestov, Some extremely amenable groups related to operator algebras and ergodic theory, J. Inst. Math. Jussieu 6 (2007), no. 2, 279–315. MR 2311665, DOI 10.1017/S1474748006000090
Eli Glasner, On minimal actions of Polish groups, Topology Appl. 85 (1998), no. 1-3, 119–125. 8th Prague Topological Symposium on General Topology and Its Relations to Modern Analysis and Algebra (1996). MR 1617456, DOI 10.1016/S0166-8641(97)00143-0
A. Gournay, Width of $\ell ^p$-balls, preprint, available online at http://front. math. ucdavis. edu/ 0711.3081, 2007.
M. Gromov and V. D. Milman, A topological application of the isoperimetric inequality, Amer. J. Math. 105 (1983), no. 4, 843–854. MR 708367, DOI 10.2307/2374298
M. Gromov, $\textrm {CAT}(\kappa )$-spaces: construction and concentration, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 280 (2001), no. Geom. i Topol. 7, 100–140, 299–300 (English, with Russian summary); English transl., J. Math. Sci. (N.Y.) 119 (2004), no. 2, 178–200. MR 1879258, DOI 10.1023/B:JOTH.0000008756.15786.0f
M. Gromov, Isoperimetry of waists and concentration of maps, Geom. Funct. Anal. 13 (2003), no. 1, 178–215. MR 1978494, DOI 10.1007/s000390300004
Misha Gromov, Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics, vol. 152, Birkhäuser Boston, Inc., Boston, MA, 1999. Based on the 1981 French original [ MR0682063 (85e:53051)]; With appendices by M. Katz, P. Pansu and S. Semmes; Translated from the French by Sean Michael Bates. MR 1699320
Misha Gromov, Topological invariants of dynamical systems and spaces of holomorphic maps. I, Math. Phys. Anal. Geom. 2 (1999), no. 4, 323–415. MR 1742309, DOI 10.1023/A:1009841100168
R. Latała and J. O. Wojtaszczyk, On the infimum convolution inequality, Studia Math. 189 (2008), no. 2, 147–187. MR 2449135, DOI 10.4064/sm189-2-5
Michel Ledoux, The concentration of measure phenomenon, Mathematical Surveys and Monographs, vol. 89, American Mathematical Society, Providence, RI, 2001. MR 1849347, DOI 10.1090/surv/089
V. D. Milman, A certain property of functions defined on infinite-dimensional manifolds, Dokl. Akad. Nauk SSSR 200 (1971), 781–784 (Russian). MR 0309150
V. D. Milman, A new proof of A. Dvoretzky’s theorem on cross-sections of convex bodies, Funkcional. Anal. i Priložen. 5 (1971), no. 4, 28–37 (Russian). MR 0293374
V. D. Milman, Asymptotic properties of functions of several variables that are defined on homogeneous spaces, Dokl. Akad. Nauk SSSR 199 (1971), 1247–1250 (Russian); English transl., Soviet Math. Dokl. 12 (1971), 1277–1281. MR 0303566
V. D. Milman, Diameter of a minimal invariant subset of equivariant Lipschitz actions on compact subsets of $\textbf {R}^k$, Geometrical aspects of functional analysis (1985/86), Lecture Notes in Math., vol. 1267, Springer, Berlin, 1987, pp. 13–20. MR 907682, DOI 10.1007/BFb0078133
V. D. Milman, The heritage of P. Lévy in geometrical functional analysis, Astérisque 157-158 (1988), 273–301. Colloque Paul Lévy sur les Processus Stochastiques (Palaiseau, 1987). MR 976223
Vitali D. Milman and Gideon Schechtman, Asymptotic theory of finite-dimensional normed spaces, Lecture Notes in Mathematics, vol. 1200, Springer-Verlag, Berlin, 1986. With an appendix by M. Gromov. MR 856576
Assaf Naor, The surface measure and cone measure on the sphere of $l_p^n$, Trans. Amer. Math. Soc. 359 (2007), no. 3, 1045–1079. MR 2262841, DOI 10.1090/S0002-9947-06-03939-0
Vladimir Pestov, Dynamics of infinite-dimensional groups and Ramsey-type phenomena, Publicações Matemáticas do IMPA. [IMPA Mathematical Publications], Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2005. 25$^\textrm {o}$ Colóquio Brasileiro de Matemática. [25th Brazilian Mathematics Colloquium]. MR 2164572
Vladimir Pestov, Ramsey-Milman phenomenon, Urysohn metric spaces, and extremely amenable groups, Israel J. Math. 127 (2002), 317–357. MR 1900705, DOI 10.1007/BF02784537
Vladimir Pestov, The isometry group of the Urysohn space as a Lev́y group, Topology Appl. 154 (2007), no. 10, 2173–2184. MR 2324929, DOI 10.1016/j.topol.2006.02.010
G. Schechtman and J. Zinn, Concentration on the $l^n_p$ ball, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 1745, Springer, Berlin, 2000, pp. 245–256. MR 1796723, DOI 10.1007/BFb0107218
Masaki Tsukamoto, Macroscopic dimension of the $l^p$-ball with respect to the $l^q$-norm, J. Math. Kyoto Univ. 48 (2008), no. 2, 445–454. MR 2436746, DOI 10.1215/kjm/1250271421