Abstract
Properties of the extreme points of families of concave measures on infinite-dimensional locally convex spaces are studied. The localization method is generalized to hyperbolic measures on Fréchet spaces.
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References
L. M. Arutyunyan and E. D. Kosov, “Polynomials on spaces with logarithmically concave measures”, Dokl. Akad. Nauk, 460:5 (2015), 503–506; English transl.:, Dokl. Math., 91:1 (2015), 72–75.
S. G. Bobkov, “Remarks on the growth of \(L^p\)-norms of polynomials”, Geometric Aspects of Functional Analysis, Proceedings of the Israel Seminar (GAFA) 1996–2000, Lecture Notes in Math, 1745 Springer-Verlag, Berlin, 2000, 27–35.
S. G. Bobkov, “Some generalizations of Prokhorov’s results on Khinchin-type inequalities for polynomials”, Teor. Veroyatnost. Primenen., 45:4 (2000), 745–748; English transl.:, Theory Probab. Appl., 45:4 (2001), 644–647.
S. G. Bobkov, “Localization proof of the isoperimetric Bakry–Ledoux inequality and some applications”, Teor. Veroyatnost. Primenen., 47:2 (2002), 340–346; English transl.:, Theory Probab. Appl., 47:2 (2003), 308–314.
S. G. Bobkov, “On isoperimetric constants for log-concave probability distributions”, Geometric Aspects of Functional Analysis, Proceedings of the Israel Seminar (GFA) 2004–2005, Lecture Notes in Math., 1910 Springer-Verlag, Berlin, 2007, 81–88.
S. G. Bobkov and J. Melbourne, “Localization for infinite-dimensional hyperbolic measures”, Dokl. Akad. Nauk, 462:3 (2015), 261–263; English transl.:, Dokl. Math., 91:3 (2015), 297–299.
S. G. Bobkov and J. Melbourne, “Hyperbolic measures on infinite dimensional spaces”, Probab. Surv., 13 (2016), 57–88.
S. G. Bobkov and F. L. Nazarov, “Sharp dilation-type inequalities with fixed parameter of convexity”, Zap. Nauchn. Sem. POMI, 351 (2007), 54–78; English transl.:, J. Math. Sci. (N. Y.), 152:6 (2008), 826–839.
V. I. Bogachev, Measure Theory. V. 1, 2, Springer, Berlin–Heidelberg, 2007.
V. I. Bogachev, Weak Convergence of Measures., Amer. Math. Soc., Providence, RI, 2018.
V. I. Bogachev and O. G. Smolyanov, Topological Vector Spaces and Their Applications, Springer, Cham, 2017.
C. Borell, “Convex measures on locally convex spaces”, Ark. Math., 12 (1974), 239–252.
C. Borell, “Convex set functions in \(d\)-space”, Period. Math. Hungar., 6:2 (1975), 111–136.
C. Borell, “Convexity of measures in certain convex cones in vector space \(\sigma \)-algebras”, Math. Scand., 53:1 (1983), 125–144.
K. Borsuk, “Drei Sätze über die \(n\)-dimensionale euklidische Sphäre”, Fund. Math., 20 (1933), 177–190.
M. Fradelizi, “Concentration inequalities for \(s\)-concave measures of dilations of Borel sets and applications”, Electron. J. Probab., 14:71 (2009), 2068–2090.
M. Fradelizi and O. Guédon, “The extreme points of subsets of \(s\)-concave probabilities and a geometric localization theorem”, Discrete Comput. Geom., 31:2 (2004), 327–335.
M. Fradelizi and O. Guédon, “A generalized localization theorem and geometric inequalities for convex bodies”, Adv. Math., 204:2 (2006), 509–529.
O. Guédon, “Kahane–Khinchine type inequalities for negative exponent”, Mathematika, 46:1 (1999), 165–173.
R. Kannan, L. Lovász, and M. Simonovits, “Isoperimetric problems for convex bodies and a localization lemma”, Discrete Comput. Geom., 13:3-4 (1995), 541–559.
L. Lovász and M. Simonovits, “Random walks in a convex body and an improved volume algorithm”, Random Struct. Algorithms, 4:4 (1993), 359–412.
J. Matoušek, Using the Borsuk–Ulam Theorem Lectures on Topological Methods in Combinatorics and Geometry, Universitext, Springer-Verlag, Berlin, 2003.
F. Nazarov, M. Sodin, and A. Vol’berg, “The geometric Kannan–Lovász–Simonovits lemma, dimension-free estimates for volumes of sublevel sets of polynomials, and distribution of zeros of random analytic functions”, Algebra Anal., 14:2 (2002), 214–234; English transl.:, St. Petersburg Math. J., 14:2 (2003), 351–366.
L. E. Payne and H. F. Weinberger, “An optimal Poincaré inequality for convex domains”, Arch. Rational Mech. Anal., :5 (1960), 286–292.
Yu. D. Burago and V. A. Zalgaller, Geometric Inequalities, Nauka (Leningrad. otd.), Leningrad, 1980.
H. Hadwiger and D. Ohmann, “Brunn–Minkowskischer Satz und Isoperimetrie”, Math. Z., 66 (1956), 1–8.
M. Gromov and V. D. Milman, “Generalization of the spherical isoperimetric inequality to uniformly convex Banach spaces”, Compositio Math., 62:3 (1987), 263–282.
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I thank the referee for useful comments.
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This work was supported by the Russian Foundation for Basic Research (grant no. 20-01-00432), by the Moscow Center of Fundamental and Applied Mathematics, and by the BASIS foundation (grant no. 18-1-6-83-1).
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Translated from Funktsional'nyi Analiz i ego Prilozheniya, 2021, Vol. 55, pp. 40-54 https://doi.org/10.4213/faa3882.
Translated by O. V. Sipacheva
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Kalinin, A.N. Localization for Hyperbolic Measures on Infinite-Dimensional Spaces. Funct Anal Its Appl 55, 286–297 (2021). https://doi.org/10.1134/S0016266321040031
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DOI: https://doi.org/10.1134/S0016266321040031