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Central and \(L^p\)-concentration of 1-Lipschitz maps into \(R\)-trees. (English) Zbl 1198.46021

Let \(\{(X_n, d_{X_n}, \mu_{X_n})\}_{n=1}^\infty\) be a sequence of metric measure spaces and \(\{(Y_n, d_{Y_n})\}_{n=1}^\infty\) be a sequence of metric spaces. We say that \(1\)-Lipschitz maps from \(\{X_n\}_n\) to \(\{Y_n\}_n\) have a concentration property if, for an arbitrary sequence \(\{f_n\}_{n=1}^\infty\) of \(1\)-Lipschitz maps \(f_n:X_n\to Y_n\), there exist points \(m_{f_n}\in Y_n\) such that \[ \mu_{X_n}(\{x\in X_n: d_{Y_n}(f_n(x),m_{f_n})\geq\varepsilon\})\to 0\text{ as }n\to \infty \] for any \(\varepsilon > 0\).
The main result of the paper is that the concentration property of \(1\)-Lipschitz maps from \(\{X_n\}\) to \(\{Y_n\}\) with each \(Y_n\) being the real line is equivalent to the concentration property of \(1\)-Lipschitz maps from \(\{X_n\}\) into \(\{Y_n\}\), with each \(Y_n\) being an \(\mathbb{R}\)-tree.
This result is proved in terms of the observable diameter. Similar results are proved for the observable central radius and the observable \(L_p\)-variation.

MSC:

46B85 Embeddings of discrete metric spaces into Banach spaces; applications in topology and computer science
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
54E35 Metric spaces, metrizability
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[1] L. Ambrosio and P. Tilli, Topics on analysis in metric spaces, Oxford Lecture Series in Mathematics and its Applications, 25 , Oxford University Press, Oxford, 2004. · Zbl 1080.28001
[2] I. Chiswell, Introduction to \(\Lambda\)-trees, World Scientific Publishing Co. Inc., River Edge, NJ, 2001. · Zbl 1004.20014
[3] K. Funano, Observable concentration of mm-spaces into nonpositively curved manifolds, preprint, available online at “http://front.math.ucdavis.edu/0701.5535”, 2007. · Zbl 1139.28006 · doi:10.1007/s10711-007-9156-6
[4] K. Funano, Observable concentration of mm-spaces into spaces with doubling measures, Geom. Dedicata, 127 (2007), 49-56. · Zbl 1139.28006 · doi:10.1007/s10711-007-9156-6
[5] M. Gromov, CAT(\(\kappa\))-spaces: construction and concentration, (Russian summary) Zap. Nauchn. Sem., S.-Peterburg, Otdel. Mat. Inst. Steklov., (POMI) 280 , Geom. i Topol., 7 (2001), 100-140, 299-300; translation in J. Math. Sci. (N. Y.) 119 (2004), 178-200.
[6] M. Gromov, Isoperimetry of waists and concentration of maps, Geom. Funct. Anal., 13 (2003), 178-215. · Zbl 1044.46057 · doi:10.1007/s000390300004
[7] M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, Based on the 1981 French original, With appendices by M. Katz, P. Pansu and S. Semmes, Translated from French by Sean Michael Bates, Progress in Mathematics, 152 , Birkhäuser Boston, Inc., Boston, MA, 1999. · Zbl 0953.53002
[8] J. Jost, Nonpositive curvature: geometric and analytic aspects, Lectures in Mathematics ETH Zörich, Birkhäuser Verlag, Basel, 1997. · Zbl 0896.53002
[9] M. Ledoux, The concentration of measure phenomenon, Mathematical Surveys and Monographs, 89 , American Mathematical Society, Providence, RI, 2001. · Zbl 0995.60002
[10] M. Ledoux and K. Oleszkiewicz, On measure concentration of vector valued maps, preprint, 2007. · Zbl 1125.60016 · doi:10.4064/ba55-3-7
[11] V. D. Milman, A new proof of A. Dvoretzky’s theorem on cross-sections of convex bodies, (Russian) Funkcional. Anal. i Priložen, 5 (1971), 28-37.
[12] V. D. Milman, The heritage of P. Lévy in geometrical functional analysis, Colloque Paul Lévy sur les Processus Stochastiques (Palaiseau, 1987), Astérisque, 157-158 (1988), 273-301. · Zbl 0681.46021
[13] V. D. Milman and G. Schechtman, Asymptotic theory of finite-dimensional normed spaces, With an appendix by M. Gromov, Lecture Notes in Mathematics, 1200 , Springer-Verlag, Berlin, 1986. · Zbl 0606.46013
[14] G. Schechtman, Concentration results and applications, Handbook of the geometry of Banach spaces, 2 , North-Holland, Amsterdam, 2003, pp. 1603-1634. · Zbl 1057.46011 · doi:10.1016/S1874-5849(03)80044-X
[15] K-T. Sturm, Probability measures on metric spaces of nonpositive curvature, Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002), Contemp. Math., 338 (2003), 357-390. · Zbl 1040.60002
[16] M. Talagrand, Concentration of measure and isoperimetric inequalities in product spaces, Inst. Hautes Études Sci. Publ. Math., 81 (1995), 73-205. · Zbl 0864.60013 · doi:10.1007/BF02699376
[17] M. Talagrand, New concentration inequalities in product spaces, Invent. Math., 126 (1996), 505-563. · Zbl 0893.60001 · doi:10.1007/s002220050108
[18] C. Villani, Topics in optimal transportation, Graduate Studies in Mathematics, 58 , American Mathematical Society, Providence, RI, 2003. · Zbl 1106.90001
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