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  • Grigoris Paouris (a1)


We study the question of whether every centred convex body K of volume 1 in ℝn has “supergaussian directions”, which means θSn−1 such that for all , where c>0 is an absolute constant. We verify that a “random” direction is indeed supergaussian for isotropic convex bodies that satisfy the hyperplane conjecture. On the other hand, we show that if, for all isotropic convex bodies, a random direction is supergaussian then the hyperplane conjecture follows.



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[1]Alonso, D., Bastero, J., Bernues, J. and Wolff, P., On the isotropy constant of projections of polytopes. J. Funct. Anal. 258 (2010), 14521465.
[2]Anttila, M., Ball, K. and Perissinaki, I., The central limit theorem for convex bodies. Trans. Amer. Math. Soc. 355 (2003), 47234735.
[3]Ball, K. M., Logarithmically concave functions and sections of convex sets in ℝn. Studia Math. 88 (1988), 6984.
[4]Ball, K. M., Normed spaces with a weak-Gordon–Lewis property. In Functional Analysis (Austin, TX, 1987/1989) (Lecture Notes in Mathematics 1470), Springer (Berlin, 1991), 3647.
[5]Berwald, L., Verallgemeinerung eines Mittelswertsatzes von J. Favard für positive konkave Funkionen. Acta Math. 79 (1947), 1737.
[6]Bobkov, S. G. and Koldobsky, A., On the central limit property of convex bodies. In Geometric Aspects of Functional Analysis (Lecture Notes in Mathematics 1807) (eds Milman, V. D. and Schechtman, G.), Springer (Berlin, 2003), 4452.
[7]Bobkov, S. G. and Nazarov, F. L., On convex bodies and log-concave probability measures with unconditional basis. In Geometric Aspects of Functional Analysis (Lecture Notes in Mathematics 1807) (eds Milman, V. D. and Schechtman, G.), Springer (Berlin, 2003), 5369.
[8]Bobkov, S. G. and Nazarov, F. L., Large deviations of typical linear functionals on a convex body with unconditional basis. In Stochastic Inequalities and Applications (Progress in Probability 56), Birkhauser (Basel, 2003), 313.
[9]Bourgain, J., On high dimensional maximal functions associated to convex bodies. Amer. J. Math. 108 (1986), 14671476.
[10]Bourgain, J., On the distribution of polynomials on high dimensional convex sets. In Geometric Aspects of Functional Analysis (Lecture Notes in Mathematics 1469) (eds Lindenstrauss, J. and Milman, V. D.), Springer (Berlin, 1991), 127137.
[11]Bourgain, J., On the isotropy constant problem for ψ 2-bodies. In Geometric Aspects of Functional Analysis (Lecture Notes in Mathematics 1807) (eds Milman, V. D. and Schechtman, G.), Springer (Berlin, 2003), 114121.
[12]Borell, C., Complements of Lyapunov’s inequality. Math. Ann. 205 (1973), 323331.
[13]Borell, C., Convex set functions in d-space. Period. Math. Hungar. 6 (1975), 111136.
[14]Brehm, U. and Voigt, J., Asymptotics of cross sections for convex bodies. Beiträge Algebra Geom. 41 (2000), 437454.
[15]Campi, S. and Gronchi, P., The L p-Busemann–Petty centroid inequality. Adv. Math. 167 (2002), 128141.
[16]Dafnis, N., Giannopoulos, A. and Guédon, O., On the isotropic constant of random polytopes. Adv. Geom. 10 (2010), 311321.
[17]Dafnis, N. and Paouris, G., Small ball probability estimates, ψ 2-behavior and the hyperplane conjecture. J. Funct. Anal. 258 (2010), 19331964.
[18]Eldan, R. and Klartag, B., Pointwise Estimates for Marginals of Convex Bodies. J. Funct. Anal. 254 (2008), 22752293.
[19]Fleury, B., Between Paouris concentration inequality and variance conjecture. Ann. Inst. Henri Poincaré Probab. Stat. 46(2) (2010), 299312.
[20]Fleury, B., Concentration in a thin Euclidean shell for log-concave measures. J. Funct. Anal. 259(4) (2010), 832841.
[21]Fleury, B., Guédon, O. and Paouris, G., A stability result for mean width of L p-centroid bodies. Adv. Math. 214 (2007), 865877.
[22]Giannopoulos, A., Notes on isotropic convex bodies. Warsaw University Notes, (2003). Available online in∼apyiannop.
[23]Giannopoulos, A., Pajor, A. and Paouris, G., A note on subgaussian estimates for linear functionals on convex bodies. Proc. Amer. Math. Soc. 135 (2007), 25992606.
[24]Giannopoulos, A., Paouris, G. and Valettas, P., On the existence of subgaussian directions for log-concave measures. In Proc. Workshop on Concentration, Functional Inequalities and Isoperimetry (Contemporary Mathematics 545), American Mathematical Society (Providence, RI, 2011), 109148.
[25]Guédon, O. and Milman, E., Interpolating thin-shell and sharp large-deviation estimates for isotropic log-concave measures. Preprint, arXiv:1011.0943.
[26]Hensley, D., Slicing convex bodies, bounds of slice area in terms of the body’s covariance. Proc. Amer. Math. Soc. 79 (1980), 619625.
[27]Junge, M., On the hyperplane conjecture for quotient spaces of L p. Forum Math. 6 (1994), 617635.
[28]Kannan, R., Lovász, L. and Simonovits, M., Isoperimetric problems for convex bodies and a localization lemma. Discrete Comput. Geom. 13 (1995), 541559.
[29]Klartag, B., On convex perturbations with a bounded isotropic constant. Geom. Funct. Anal. 16 (2006), 12741290.
[30]Klartag, B., A central limit theorem for convex sets. Invent. Math. 168 (2007), 91131.
[31]Klartag, B., Power-law estimates for the central limit theorem for convex sets. J. Funct. Anal. 245 (2007), 284310.
[32]Klartag, B., Uniform almost sub-gaussian estimates for linear functionals on convex sets. Algebra i Analiz (St. Petersburg Math. J.) 19(1) (2007), 109148.
[33]Klartag, B., A Berry-Esseen type inequality for convex bodies with an unconditional basis. Probab. Theory Related Fields 45 (2009), 133.
[34]Klartag, B., On nearly radial marginals of high-dimensional probability measures. J. Eur. Math. Soc. (JEMS) 12 (2010), 723754.
[35]Klartag, B. and Kozma, G., On the hyperplane conjecture for random convex sets. Israel J. Math. 170 (2009), 253268.
[36]Klartag, B. and Milman, E., Centroid bodies and the logarithmic Laplace transform—a unified approach. Preprint, arXiv:1103.2985v1.
[37]Klartag, B. and Vershynin, R., Small ball probability and Dvoretzky Theorem. Israel J. Math. 157 (2007), 193207.
[38]Koldobsky, A., Pajor, A. and Yaskin, V., Inequalities of the Kahane–Khinchin type and sections of L p-balls. Studia Math. 184(3) (2008), 217231.
[39]König, H., Meyer, M. and Pajor, A., The isotropy constants of the Schatten classes are bounded. Math. Ann. 312 (1998), 773783.
[40]Latala, R. and Oleszkiewicz, K., Small ball probability estimates in terms of width. Studia Math. 169 (2005), 305314.
[41]Latala, R. and Wojtaszczyk, J. O., On the infimum convolution inequality. Studia Math. 189 (2008), 147187.
[42]Litvak, A., Milman, V. D. and Schechtman, G., Averages of norms and quasi-norms. Math. Ann. 312 (1998), 95124.
[43]Lutwak, E., Yang, D. and Zhang, G., L p affine isoperimetric inequalities. J. Differential Geom. 56 (2000), 111132.
[44]Lutwak, E. and Zhang, G., Blaschke–Santaló inequalities. J. Differential Geom. 47 (1997), 116.
[45]Milman, E., On the role of convexity in isoperimetry, spectral gap and concentration. Invent. Math. 177 (2009), 143.
[46]Milman, V. D., A new proof of A. Dvoretzky’s theorem in cross-sections of convex bodies. Funktsional. Anal. i Prilozen. 5 (1971), 2837 (in Russian).
[47]Milman, V. D. and Pajor, A., Isotropic positions and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space. In GAFA Seminar 87–89 (Lecture Notes in Mathematics 1376) , Springer (1989), 64–104.
[48]Milman, V. D. and Schechtman, G., Asymptotic Theory of Finite Dimensional Normed Spaces (Lecture Notes in Mathematics 1200), Springer (Berlin, 1986).
[49]Milman, V. D. and Schechtman, G., Global versus Local asymptotic theories of finite-dimensional normed spaces. Duke Math. J. 90 (1997), 7393.
[50]Paouris, G., Ψ2-estimates for linear functionals on zonoids. In Geometric Aspects of Functional Analysis (Lecture Notes in Mathematics 1807) (eds Milman, V. D. and Schechtman, G.), Springer (Berlin, 2003), 211222.
[51]Paouris, G., Concentration of mass on convex bodies. Geom. Funct. Anal. 16 (2006), 10211049.
[52]Paouris, G., Small ball probability estimates for log-concave measures. Trans. Amer. Math. Soc. (2011), doi:10.1090/S0002-9947-2011-05411-5.
[53]Pisier, G., The Volume of Convex Bodies and Banach Space Geometry (Cambridge Tracts in Mathematics 94), Cambridge University Press (Cambridge, 1989).
[54]Pivovarov, P., On the volume of caps and bounding the mean-width of an isotropic convex body. Math. Proc. Cambridge Philos. Soc. 149(2) (2010), 317331.
[55]Schneider, R., Convex Bodies: the Brunn–Minkowski Theory (Encyclopedia of Mathematics and its Applications 44), Cambridge University Press (Cambridge, 1993).
[56]Sodin, S., An isoperimetric inequality on the np balls. Ann. Inst. Henri Poincaré Probab. Stat. 44 (2008), 362373.
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