Abstract
We introduce and initiate the study of new parameters associated with any norm and any log-concave measure on ℝn, which provide sharp distributional inequalities. In the Gaussian context this investigation sheds light to the importance of the statistical measures of dispersion of the norm in connection with the local structure of the ambient space. As a byproduct of our study, we provide a short proof of Dvoretzky’s theorem which not only supports the aforementioned significance but also complements the classical probabilistic formulation.
Communicated by: M. Henk
Funding The first author was supported by the NSF CAREER-1151711 grant. The second author was supported by the NSF grant DMS-1612936.
Acknowledgement
The authors are grateful to Peter Pivovarov for useful remarks. They would also like to thank the anonymous referee whose helpful comments improved the presentation of the paper.
References
[1] R. Adamczak, C. Strzeleki, On the convex Poincaré inequality and weak transportation transportation inequalities. Bernoulli25 (2019), 341–374. MR3892322 Zbl 0700721010.3150/17-BEJ989Search in Google Scholar
[2] F. Albiac, N. J. Kalton, Topics in Banach space theory. Springer 2016. MR3526021 Zbl 1352.4600210.1007/978-3-319-31557-7Search in Google Scholar
[3] S. Artstein-Avidan, A. Giannopoulos, V. D. Milman, Asymptotic geometric analysis. Part I, volume 202 of Mathematical Surveys and Monographs. Amer. Math. Soc. 2015. MR3331351 Zbl 1337.5200110.1090/surv/202Search in Google Scholar
[4] S. G. Bobkov, Isoperimetric and analytic inequalities for log-concave probability measures. Ann. Probab. 27 (1999), 1903–1921. MR1742893 Zbl 0964.6001310.1214/aop/1022874820Search in Google Scholar
[5] S. G. Bobkov, F. Götze, Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163 (1999), 1–28. MR1682772 Zbl 0924.4602710.1006/jfan.1998.3326Search in Google Scholar
[6] S. G. Bobkov, M. Ledoux, From Brunn–Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities. Geom. Funct. Anal. 10 (2000), 1028–1052. MR1800062 Zbl 0969.2601910.1007/PL00001645Search in Google Scholar
[7] V. I. Bogachev, Gaussian measures, volume 62 of Mathematical Surveys and Monographs. Amer. Math. Soc. 1998. MR1642391 Zbl 0913.6003510.1090/surv/062Search in Google Scholar
[8] C. Borell, Convex set functions in d-space. Period. Math. Hungar. 6 (1975), 111–136. MR0404559 Zbl 0307.2800910.1007/BF02018814Search in Google Scholar
[9] S. Boucheron, G. Lugosi, P. Massart, Concentration inequalities. Oxford Univ. Press 2013. MR3185193 Zbl 1279.6000510.1093/acprof:oso/9780199535255.001.0001Search in Google Scholar
[10] S. Chatterjee, Superconcentration and related topics. Springer 2014. MR3157205 Zbl 1288.6000110.1007/978-3-319-03886-5Search in Google Scholar
[11] D. Cordero-Erausquin, M. Fradelizi, B. Maurey, The (B) conjecture for the Gaussian measure of dilates of symmetric convex sets and related problems. J. Funct. Anal. 214 (2004), 410–427. MR2083308 Zbl 1073.6004210.1016/j.jfa.2003.12.001Search in Google Scholar
[12] D. Cordero-Erausquin, M. Ledoux, Hypercontractive measures, Talagrand’s inequality, and influences. In: Geometric aspects of functional analysis, volume 2050 of Lecture Notes in Math., 169–189, Springer 2012. MR2985132 Zbl 1280.6001810.1007/978-3-642-29849-3_10Search in Google Scholar
[13] A. Dvoretzky, Some results on convex bodies and Banach spaces. In: Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960), 123–160, Academic Press 1961. MR0139079 Zbl 0119.31803Search in Google Scholar
[14] A. Dvoretzky, C. A. Rogers, Absolute and unconditional convergence in normed linear spaces. Proc. Nat. Acad. Sci. U. S. A. 36 (1950), 192–197. MR0033975 Zbl 0036.3630310.1073/pnas.36.3.192Search in Google Scholar PubMed PubMed Central
[15] A. Ehrhard, Symétrisation dans ľespace de Gauss. Math. Scand. 53 (1983), 281–301. MR745081 Zbl 0542.6000310.7146/math.scand.a-12035Search in Google Scholar
[16] A. Eskenazis, T. Tkocz, Gaussian mixtures: entropy and geometric inequalities. Ann. Probab. 46 (2018), 2908–2945. MR3846841 Zbl 0696435110.1214/17-AOP1242Search in Google Scholar
[17] T. Figiel, A short proof of Dvoretzky’s theorem on almost spherical sections of convex bodies. Compositio Math. 33 (1976), 297–301. MR0487392 Zbl 0343.52004Search in Google Scholar
[18] T. Figiel, J. Lindenstrauss, V. D. Milman, The dimension of almost spherical sections of convex bodies. Acta Math. 139 (1977), 53–94. MR0445274 Zbl 0375.5200210.1007/BF02392234Search in Google Scholar
[19] Y. Gordon, Some inequalities for Gaussian processes and applications. Israel J. Math. 50 (1985), 265–289. MR800188 Zbl 0663.6003410.1007/BF02759761Search in Google Scholar
[20] M. Gromov, V. D. Milman, A topological application of the isoperimetric inequality. Amer. J. Math. 105 (1983), 843–854. MR708367 Zbl 0522.5303910.2307/2374298Search in Google Scholar
[21] H. Huang, F. Wei, Upper bound for the Dvoretzky dimension in Milman-Schechtman theorem. In: Geometric aspects of functional analysis, volume 2169 of Lecture Notes in Math., 181–186, Springer 2017. MR3645122 Zbl 1369.5200410.1007/978-3-319-45282-1_12Search in Google Scholar
[22] F. John, Extremum problems with inequalities as subsidiary conditions. In: Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, 187–204, Interscience Publ. 1948. MR0030135 Zbl 0034.10503Search in Google Scholar
[23] W. B. Johnson, J. Lindenstrauss, Extensions of Lipschitz mappings into a Hilbert space. In: Conference in modern analysis and probability (New Haven, Conn., 1982), volume 26 of Contemp. Math., 189–206, Amer. Math. Soc. 1984. MR737400 Zbl 0539.4601710.1090/conm/026/737400Search in Google Scholar
[24] B. Klartag, A geometric inequality and a low M-estimate. Proc. Amer. Math. Soc. 132 (2004), 2619–2628. MR2054787 Zbl 1064.5200310.1090/S0002-9939-04-07484-2Search in Google Scholar
[25] B. Klartag, Uniform almost sub-Gaussian estimates for linear functionals on convex sets. (Russian) Algebra i Analiz19 (2007), 109–148. English translation: St. Petersburg Math. J. 19 (2008), 77–106. MR2319512 Zbl 1140.60010Search in Google Scholar
[26] B. Klartag, R. Vershynin, Small ball probability and Dvoretzky’s theorem. Israel J. Math. 157 (2007), 193–207. MR2342445 Zbl 1120.4600310.1007/s11856-006-0007-1Search in Google Scholar
[27] R. Latała, K. Oleszkiewicz, Small ball probability estimates in terms of widths. Studia Math. 169 (2005), 305–314. MR2140804 Zbl 1120.4600310.4064/sm169-3-6Search in Google Scholar
[28] M. Ledoux, The concentration of measure phenomenon, volume 89 of Mathematical Surveys and Monographs. Amer. Math. Soc. 2001. MR1849347 Zbl 0995.60002Search in Google Scholar
[29] M. Ledoux, M. Talagrand, Probability in Banach spaces. Springer 2011. MR2814399 Zbl 1226.60003Search in Google Scholar
[30] D. R. Lewis, Finite dimensional subspaces of Lp. Studia Math. 63 (1978), 207–212. MR511305 Zbl 0406.4602310.4064/sm-63-2-207-212Search in Google Scholar
[31] E. Meckes, Concentration of Measure on the Compact Classical Matrix Groups, Lecture notes 2014. www.case.edu/artsci/math/esmeckes/Haar_notes.pdfSearch in Google Scholar
[32] V. D. Milman, A new proof of A. Dvoretzky’s theorem on cross-sections of convex bodies. (Russian) Funkcional. Anal. i Priložen. 5 (1971), no. 4, 28–37. English translation: Functional Anal. Appl. 5 (1971), 288–295. MR0293374 Zbl 0239.46018Search in Google Scholar
[33] V. D. Milman, A few observations on the connections between local theory and some other fields. In: Geometric aspects of functional analysis (1986/87), volume 1317 of Lecture Notes in Math., 283–289, Springer 1988. MR950988 Zbl 0657.1002010.1007/BFb0081748Search in Google Scholar
[34] V. D. Milman, A. Pajor, Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space. In: Geometric aspects of functional analysis (1987–88), volume 1376 of Lecture Notes in Math., 64–104, Springer 1989. MR1008717 Zbl 0679.4601210.1007/BFb0090049Search in Google Scholar
[35] V. D. Milman, G. Schechtman, Asymptotic theory of finite-dimensional normed spaces. Springer 1986. MR856576 Zbl 0606.46013Search in Google Scholar
[36] V. D. Milman, G. Schechtman, Global versus local asymptotic theories of finite-dimensional normed spaces. Duke Math. J. 90 (1997), 73–93. MR1478544 Zbl 0911.5200210.1215/S0012-7094-97-09003-7Search in Google Scholar
[37] V. D. Milman, S. J. Szarek, A geometric lemma and duality of entropy numbers. In: Geometric aspects of functional analysis, volume 1745 of Lecture Notes in Math., 191–222, Springer 2000. MR1796720 Zbl 0987.4601710.1007/BFb0107215Search in Google Scholar
[38] A. Naor, D. Romik, Projecting the surface measure of the sphere of
[39] F. Otto, C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173 (2000), 361–400. MR1760620 Zbl 0985.5801910.1006/jfan.1999.3557Search in Google Scholar
[40] G. Paouris, P. Pivovarov, J. Zinn, A central limit theorem for projections of the cube. Probab. Theory Related Fields159 (2014), 701–719. MR3230006 Zbl 1301.5201710.1007/s00440-013-0518-8Search in Google Scholar
[41] G. Paouris, P. Valettas, A Gaussian small deviation inequality for convex functions. Annals of Probability46 (2018) 1441–1454. MR3785592 Zbl 0689477810.1214/17-AOP1206Search in Google Scholar
[42] G. Paouris, P. Valettas, On Dvoretzky’s theorem for subspaces of Lp. J. Funct. Anal. 275 (2018) 2225–2252. MR3841541 Zbl 0691570510.1016/j.jfa.2018.07.008Search in Google Scholar
[43] G. Paouris, P. Valettas, J. Zinn, Random version of Dvoretzky’s theorem in
[44] G. Pisier, Probabilistic methods in the geometry of Banach spaces. In: Probability and analysis (Varenna, 1985), volume 1206 of Lecture Notes in Math., 167–241, Springer 1986. MR864714 Zbl 0606.6000810.1007/BFb0076302Search in Google Scholar
[45] G. Pisier, The volume of convex bodies and Banach space geometry, volume 94 of Cambridge Tracts in Mathematics. Cambridge Univ. Press 1989. MR1036275 Zbl 0698.4600810.1017/CBO9780511662454Search in Google Scholar
[46] P.-M. Samson, Concentration inequalities for convex functions on product spaces. In: Stochastic inequalities and applications, volume 56 of Progr. Probab., 33–52, Birkhäuser 2003. MR2073425 Zbl 1037.6001910.1007/978-3-0348-8069-5_4Search in Google Scholar
[47] G. Schechtman, A remark concerning the dependence on ε in Dvoretzky’s theorem. In: Geometric aspects of functional analysis (1987–88), volume 1376 of Lecture Notes in Math., 274–277, Springer 1989. MR1008729 Zbl 0679.4601110.1007/BFb0090061Search in Google Scholar
[48] G. Schechtman, The random version of Dvoretzky’s theorem in
[49] G. Schechtman, J. Zinn, On the volume of the intersection of two
[50] A. Szankowski, On Dvoretzky’s theorem on almost spherical sections of convex bodies. Israel J. Math. 17 (1974), 325–338. MR0350388 Zbl 0288.5200210.1007/BF02756881Search in Google Scholar
[51] M. Talagrand, On Russo’s approximate zero-one law. Ann. Probab. 22 (1994), 1576–1587. MR1303654 Zbl 0819.2800210.1214/aop/1176988612Search in Google Scholar
[52] M. Talagrand, Transportation cost for Gaussian and other product measures. Geom. Funct. Anal. 6 (1996), 587–600. MR1392331 Zbl 0859.4603010.1007/BF02249265Search in Google Scholar
[53] K. Tikhomirov, Superconcentration, and randomized Dvoretzky’s theorem for spaces with 1-unconditional bases. J. Funct. Anal. 274 (2018), 121–151. MR3718050 Zbl 0680273310.1016/j.jfa.2017.08.021Search in Google Scholar
[54] K. E. Tikhomirov, The randomized Dvoretzky’s theorem in
[55] N. Tomczak-Jaegermann, Banach–Mazur distances and finite-dimensional operator ideals, volume 38 of Pitman Monographs and Surveys in Pure and Applied Mathematics. Longman 1989. MR993774 Zbl 0721.46004Search in Google Scholar
[56] P. Valettas, On the tightness of Gaussian concentration for convex functions. Journal d’Analyse Mathematique, to appear. arXiv:1706.09446 [math.PR]10.1007/s11854-021-0073-7Search in Google Scholar
© 2019 Walter de Gruyter GmbH, Berlin/Boston
