Hostname: page-component-848d4c4894-5nwft Total loading time: 0 Render date: 2024-05-25T18:47:42.440Z Has data issue: false hasContentIssue false
  • English
  • Français

On the volume of caps and bounding the mean-width of an isotropic convex body

Published online by Cambridge University Press:  10 May 2010

PETER PIVOVAROV
PETER PIVOVAROV*
Affiliation:
Department of Mathematical and Statistical Sciences, 632 CAB, University of Alberta, Edmonton, AB, Canada, T6G-2G1. e-mail: ppivovarov@math.ualberta.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

Let K be a convex body which is (i) symmetric with respect to each of the coordinate hyperplanes and (ii) in isotropic position. We prove that most linear functionals acting on K exhibit super-Gaussian tail behavior. Using known facts about the mean-width of such bodies, we then deduce strong lower bounds for the volume of certain caps. We also prove a converse statement. Namely, if an arbitrary isotropic convex body (not necessarily satisfying (i)) exhibits similar cap-behavior, then one can bound its mean-width.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 149 , Issue 2 , September 2010 , pp. 317 - 331
Copyright
Copyright © Cambridge Philosophical Society 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Ball, K. An elementary introduction to modern convex geometry. In Flavors of geometry, Math. Sci. Res. Inst. Publ. vol. 31 (Cambridge University Press, 1997), pp. 1–58.Google Scholar
[2]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. Progr. Probab. vol. 56, (Birkhäuser, 2003), pp. 3–13.CrossRefGoogle Scholar
[3]Bobkov, S. G. and Nazarov, F. L. On convex bodies and log-concave probability measures with unconditional basis. In Geometric Aspects of Functional Analysis. vol. 1807 Lecture Notes in Math. (Springer, 2003), pp. 53–69.CrossRefGoogle Scholar
[4]Bourgain, J. On the distribution of polynomials on high-dimensional convex sets. In Geometric Aspects of Functional Analysis (1989–90), Lecture Notes in Math. vol. 1469 (Springer, 1991), pp. 127–137.CrossRefGoogle Scholar
[5]Dafnis, N., Giannopoulos, A. and Tsolomitis, A.Asymptotic shape of a random polytope in a convex body. J. Funct. Anal. 257 (9) (2009), 2820–2839.CrossRefGoogle Scholar
[6]Figiel, T. and Tomczak–Jaegermann, N.Projections onto Hilbertian subspaces of Banach spaces. Israel J. Math. 33 (2) (1979), 155–171.CrossRefGoogle Scholar
[7]Giannopoulos, A., Pajor, A. and Paouris, G.A note on subgaussian estimates for linear functionals on convex bodies. Proc. Amer. Math. Soc. 135 (8) (2007), 2599–2606 (electronic).CrossRefGoogle Scholar
[8]Giannopoulos, A. A. Notes on isotropic convex bodies (2003). available at http://users.uoa.gr/~apgiannop/.Google Scholar
[9]Giannopoulos, A. A. and Milman, V. D.Mean width and diameter of proportional sections of a symmetric convex body. J. Reine Angew. Math. 497 (1998), 113–139.CrossRefGoogle Scholar
[10]Giannopoulos, A. A. and Milman, V. D.Concentration property on probability spaces. Adv. Math. 156 (1) (2000), 77–106.CrossRefGoogle Scholar
[11]Giannopoulos, A. A. and Milman, V. D. Asymptotic convex geometry: short overview. In Different Faces of Geometry, Int. Math. Ser. (N. Y.) vol. 3 (Kluwer/Plenum, 2004), pp. 87–162.CrossRefGoogle Scholar
[12]Hartzoulaki, M.Probabilistic methods in the theory of convex bodies. PhD thesis, University of Crete (2003).Google Scholar
[13]Kannan, R., Lovász, L. and Simonovits, M.Isoperimetric problems for convex bodies and a localization lemma. Discrete Comput. Geom. 13 (3-4) (1995), 541–559.CrossRefGoogle Scholar
[14]Klartag, B. On nearly radial marginals of high-dimensional probability measures. to appear, J. Eur. Math. Soc. Available at http://www.math.tau.ac.il/~klartagb/publications.html.Google Scholar
[15]Klartag, B.Uniform almost sub-Gaussian estimates for linear functionals on convex sets. Algebra i Analiz 19 (1) (2007), 109–148.Google Scholar
[16]Kwapień, S. and Woyczyński, W. A.Random series and stochastic integrals: single and multiple. Probab. Appl. (Birkhäuser Boston Inc. 1992).CrossRefGoogle Scholar
[17]Ledoux, M. and Talagrand, M.Probability in Banach spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. vol. 23 (Springer-Verlag, 1991). Isoperimetry and processes.CrossRefGoogle Scholar
[18]Litvak, A. E., Milman, V. D. and Pajor, A.The covering numbers and “low M*-estimate” for quasi-convex bodies. Proc. Amer. Math. Soc. 127 (5) (1999), 1499–1507.CrossRefGoogle Scholar
[19]Milman, V. D. and Pajor, A. Isotropic position, inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space. In Geometric Aspects of Functional Analysis (1987–88), Lecture Notes in Math. vol. 1376 (Springer, 1989), pp. 64–104.CrossRefGoogle Scholar
[20]Milman, V. D. and Schechtman, G.Asymptotic theory of finite-dimensional normed spaces, Lecture Notes in Mathematics. vol. 1200 (Springer-Verlag, 1986). With an appendix by M. Gromov.Google Scholar
[21]Paouris, G. Super-Gaussian estimates for log-concave measures. In preparation.Google Scholar
[22]Paouris, G. Ψ2-estimates for linear functionals on zonoids. In Geometric Aspects of Functional Analysis Lecture Notes in Math. vol. 1807 (Springer, 2003), pp. 211–222.CrossRefGoogle Scholar
[23]Paouris, G.Concentration of mass on convex bodies. Geom. Funct. Anal. 16 (5) (2006), 1021–1049.CrossRefGoogle Scholar
[24]Pisier, G.Holomorphic semigroups and the geometry of Banach spaces. Ann. of Math. (2), 115 (2) (1982), 375–392.CrossRefGoogle Scholar
[25]Pisier, G.The volume of convex bodies and Banach space geometry. Cambridge Tracts in Mathematics. vol. 94 (Cambridge University Press, 1989).CrossRefGoogle Scholar
[26]Rudelson, M. and Vershynin, R.The Littlewood-Offord problem and invertibility of random matrices. Adv. Math. 218 (2) (2008), 600–633.CrossRefGoogle Scholar
[27]Schmuckenschläger, M. A concentration of measure phenomenon on uniformly convex bodies. In Geometric Aspects of Functional Analysis (Israel, 1992–1994), Oper. Theory Adv. Appl. vol. 77 (Birkhäuser, 1995), pp. 275–287.CrossRefGoogle Scholar