Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A note on subgaussian estimates for linear functionals on convex bodies


Authors: A. Giannopoulos, A. Pajor and G. Paouris
Journal: Proc. Amer. Math. Soc. 135 (2007), 2599-2606
MSC (2000): Primary 52A20; Secondary 46B07
Published electronically: March 29, 2007
MathSciNet review: 2302581
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We give an alternative proof of a recent result of Klartag on the existence of almost subgaussian linear functionals on convex bodies. If $ K$ is a convex body in $ {\mathbb{R}}^n$ with volume one and center of mass at the origin, there exists $ x\neq 0$ such that

$\displaystyle \vert\{ y\in K:\,\vert\langle y,x\rangle \vert\geq t\Vert\langle\cdot ,x\rangle\Vert _1\}\vert\leq\exp (-ct^2/\log^2(t+1))$

for all $ t\geq 1$, where $ c>0$ is an absolute constant. The proof is based on the study of the $ L_q$-centroid bodies of $ K$. Analogous results hold true for general log-concave measures.


References [Enhancements On Off] (What's this?)

  • 1. Keith Ball, Logarithmically concave functions and sections of convex sets in 𝑅ⁿ, Studia Math. 88 (1988), no. 1, 69–84. MR 932007
  • 2. S. G. Bobkov and F. L. Nazarov, On convex bodies and log-concave probability measures with unconditional basis, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 1807, Springer, Berlin, 2003, pp. 53–69. MR 2083388, 10.1007/978-3-540-36428-3_6
  • 3. Sergey G. Bobkov and Fedor L. Nazarov, Large deviations of typical linear functionals on a convex body with unconditional basis, Stochastic inequalities and applications, Progr. Probab., vol. 56, Birkhäuser, Basel, 2003, pp. 3–13. MR 2073422
  • 4. J. Bourgain, On the distribution of polynomials on high-dimensional convex sets, Geometric aspects of functional analysis (1989–90), Lecture Notes in Math., vol. 1469, Springer, Berlin, 1991, pp. 127–137. MR 1122617, 10.1007/BFb0089219
  • 5. J. Bourgain, On the isotropy-constant problem for “PSI-2”-bodies, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 1807, Springer, Berlin, 2003, pp. 114–121. MR 2083391, 10.1007/978-3-540-36428-3_9
  • 6. J. Bourgain, B. Klartag, and V. Milman, Symmetrization and isotropic constants of convex bodies, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 1850, Springer, Berlin, 2004, pp. 101–115. MR 2087154, 10.1007/978-3-540-44489-3_10
  • 7. S. Campi and P. Gronchi, The 𝐿^{𝑝}-Busemann-Petty centroid inequality, Adv. Math. 167 (2002), no. 1, 128–141. MR 1901248, 10.1006/aima.2001.2036
  • 8. B. Klartag, On convex perturbations with a bounded isotropic constant, Geom. Funct. Anal. (2006), to appear.
  • 9. B. Klartag, Uniform almost sub-gaussian estimates for linear functionals on convex sets, Preprint.
  • 10. Erwin Lutwak and Gaoyong Zhang, Blaschke-Santaló inequalities, J. Differential Geom. 47 (1997), no. 1, 1–16. MR 1601426
  • 11. Erwin Lutwak, Deane Yang, and Gaoyong Zhang, 𝐿_{𝑝} affine isoperimetric inequalities, J. Differential Geom. 56 (2000), no. 1, 111–132. MR 1863023
  • 12. 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, 10.1007/BFb0090049
  • 13. Vitali D. Milman and Gideon Schechtman, Asymptotic theory of finite-dimensional normed spaces, Lecture Notes in Mathematics, vol. 1200, Springer-Verlag, Berlin, 1986. With an appendix by M. Gromov. MR 856576
  • 14. G. Paouris, Ψ₂-estimates for linear functionals on zonoids, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 1807, Springer, Berlin, 2003, pp. 211–222. MR 2083399, 10.1007/978-3-540-36428-3_17
  • 15. G. Paouris, On the 𝜓₂-behaviour of linear functionals on isotropic convex bodies, Studia Math. 168 (2005), no. 3, 285–299. MR 2146128, 10.4064/sm168-3-7
  • 16. G. Paouris, Concentration of mass on convex bodies, Geom. Funct. Anal. (2006), to appear.
  • 17. Gilles Pisier, The volume of convex bodies and Banach space geometry, Cambridge Tracts in Mathematics, vol. 94, Cambridge University Press, Cambridge, 1989. MR 1036275
  • 18. Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. MR 1216521

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 52A20, 46B07

Retrieve articles in all journals with MSC (2000): 52A20, 46B07


Additional Information

A. Giannopoulos
Affiliation: Department of Mathematics, University of Athens, Panepistimiopolis 157 84, Athens, Greece
Email: apgiannop@math.uoa.gr

A. Pajor
Affiliation: Équipe d’Analyse et de Mathématiques Appliquées, Université de Marne-la-Vallée, Champs sur Marne, 77454, Marne-la-Vallée, Cedex 2, France
Email: Alain.Pajor@univ-mlv.fr

G. Paouris
Affiliation: Équipe d’Analyse et de Mathématiques Appliquées, Université de Marne-la-Vallée, Champs sur Marne, 77454, Marne-la-Vallée, Cedex 2, France
Email: grigoris_paouris@yahoo.co.uk

DOI: https://doi.org/10.1090/S0002-9939-07-08778-3
Keywords: Isotropic convex bodies, concentration of volume, tail estimates for linear functionals, $L_q$--centroid bodies
Received by editor(s): April 20, 2006
Published electronically: March 29, 2007
Additional Notes: The project was co-funded by the European Social Fund and National Resources - (EPEAEK II) “Pythagoras II". The second named author would like to thank the Department of Mathematics of the University of Athens for the hospitality. The third named author was supported by a Marie Curie Intra-European Fellowship (EIF), Contract MEIF-CT-2005-025017.
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2007 American Mathematical Society