$\psi _{\alpha }$estimates for marginals of logconcave probability measures
HTML articles powered by AMS MathViewer
 by A. Giannopoulos, G. Paouris and P. Valettas PDF
 Proc. Amer. Math. Soc. 140 (2012), 12971308 Request permission
Abstract:
We show that a random marginal $\pi _F(\mu )$ of an isotropic logconcave probability measure $\mu$ on $\mathbb R^n$ exhibits better $\psi _{\alpha }$behavior. For a natural variant $\psi _{\alpha }^{\prime }$ of the standard $\psi _{\alpha }$norm we show the following:

[(i)] If $k\leq \sqrt {n}$, then for a random $F\in G_{n,k}$ we have that $\pi _F(\mu )$ is a $\psi _2^{\prime }$measure. We complement this result by showing that a random $\pi _F(\mu )$ is, at the same time, superGaussian.

[(ii)] If $k=n^{\delta }$, $\frac {1}{2}<\delta <1$, then for a random $F\in G_{n,k}$ we have that $\pi _F(\mu )$ is a $\psi _{\alpha (\delta )}^{\prime }$measure, where $\alpha (\delta )=\frac {2\delta }{3\delta 1}$.
References
 Keith Ball, Logarithmically concave functions and sections of convex sets in $\textbf {R}^n$, Studia Math. 88 (1988), no. 1, 69–84. MR 932007, DOI 10.4064/sm8816984
 C. Borell, Convex set functions in $d$space, Period. Math. Hungar. 6 (1975), no. 2, 111–136. MR 404559, DOI 10.1007/BF02018814
 Nikos Dafnis and Grigoris Paouris, Small ball probability estimates, $\psi _2$behavior and the hyperplane conjecture, J. Funct. Anal. 258 (2010), no. 6, 1933–1964. MR 2578460, DOI 10.1016/j.jfa.2009.06.038
 B. Fleury, Concentration in a thin Euclidean shell for logconcave measures, J. Funct. Anal. 259 (2010), no. 4, 832–841. MR 2652173, DOI 10.1016/j.jfa.2010.04.019
 B. Fleury, O. Guédon, and G. Paouris, A stability result for mean width of $L_p$centroid bodies, Adv. Math. 214 (2007), no. 2, 865–877. MR 2349721, DOI 10.1016/j.aim.2007.03.008
 A. Giannopoulos, Notes on isotropic convex bodies, Warsaw University Notes (2003).
 B. Klartag, A central limit theorem for convex sets, Invent. Math. 168 (2007), no. 1, 91–131. MR 2285748, DOI 10.1007/s0022200600288
 B. Klartag, Powerlaw estimates for the central limit theorem for convex sets, J. Funct. Anal. 245 (2007), no. 1, 284–310. MR 2311626, DOI 10.1016/j.jfa.2006.12.005
 Bo’az Klartag, On nearly radial marginals of highdimensional probability measures, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 3, 723–754. MR 2639317, DOI 10.4171/JEMS/213
 A. E. Litvak, V. D. Milman, and G. Schechtman, Averages of norms and quasinorms, Math. Ann. 312 (1998), no. 1, 95–124. MR 1645952, DOI 10.1007/s002080050213
 Erwin Lutwak and Gaoyong Zhang, BlaschkeSantaló inequalities, J. Differential Geom. 47 (1997), no. 1, 1–16. MR 1601426
 Erwin Lutwak, Deane Yang, and Gaoyong Zhang, $L_p$ affine isoperimetric inequalities, J. Differential Geom. 56 (2000), no. 1, 111–132. MR 1863023
 V. D. Milman, A new proof of A. Dvoretzky’s theorem on crosssections of convex bodies, Funkcional. Anal. i Priložen. 5 (1971), no. 4, 28–37 (Russian). MR 0293374
 V. D. Milman and A. Pajor, Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed $n$dimensional space, Geometric aspects of functional analysis (1987–88), Lecture Notes in Math., vol. 1376, Springer, Berlin, 1989, pp. 64–104. MR 1008717, DOI 10.1007/BFb0090049
 Vitali D. Milman and Gideon Schechtman, Asymptotic theory of finitedimensional normed spaces, Lecture Notes in Mathematics, vol. 1200, SpringerVerlag, Berlin, 1986. With an appendix by M. Gromov. MR 856576
 V. D. Milman and G. Schechtman, Global versus local asymptotic theories of finitedimensional normed spaces, Duke Math. J. 90 (1997), no. 1, 73–93. MR 1478544, DOI 10.1215/S0012709497090037
 G. Paouris, Concentration of mass on convex bodies, Geom. Funct. Anal. 16 (2006), no. 5, 1021–1049. MR 2276533, DOI 10.1007/s0003900605845
 G. Paouris, Small ball probability estimates for logconcave measures, Trans. Amer. Math. Soc. (to appear).
 G. Paouris, On the existence of supergaussian directions on convex bodies, preprint.
 Gilles Pisier, The volume of convex bodies and Banach space geometry, Cambridge Tracts in Mathematics, vol. 94, Cambridge University Press, Cambridge, 1989. MR 1036275, DOI 10.1017/CBO9780511662454
 Peter Pivovarov, On the volume of caps and bounding the meanwidth of an isotropic convex body, Math. Proc. Cambridge Philos. Soc. 149 (2010), no. 2, 317–331. MR 2670218, DOI 10.1017/S0305004110000216
Additional Information
 A. Giannopoulos
 Affiliation: Department of Mathematics, University of Athens, Panepistimioupolis 157 84, Athens, Greece
 Email: apgiannop@math.uoa.gr
 G. Paouris
 Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
 MR Author ID: 671202
 Email: grigoris_paouris@yahoo.co.uk
 P. Valettas
 Affiliation: Department of Mathematics, University of Athens, Panepistimioupolis 157 84, Athens, Greece
 MR Author ID: 957443
 Email: petvalet@math.uoa.gr
 Received by editor(s): July 27, 2010
 Received by editor(s) in revised form: December 24, 2010
 Published electronically: August 3, 2011
 Additional Notes: The second author was partially supported by an NSF grant
 Communicated by: Thomas Schlumprecht
 © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.  Journal: Proc. Amer. Math. Soc. 140 (2012), 12971308
 MSC (2010): Primary 46B07; Secondary 52A20
 DOI: https://doi.org/10.1090/S000299392011109845
 MathSciNet review: 2869113