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)

 
 

 

The covering numbers and ``low $M^*$-estimate"
for quasi-convex bodies


Authors: A. E. Litvak, V. D. Milman and A. Pajor
Journal: Proc. Amer. Math. Soc. 127 (1999), 1499-1507
MSC (1991): Primary 52C17; Secondary 46B07, 52A21, 52A30
DOI: https://doi.org/10.1090/S0002-9939-99-04593-1
Published electronically: January 29, 1999
MathSciNet review: 1469422
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This article gives estimates on the covering numbers and diameters of random proportional sections and projections of quasi-convex bodies in $\mathbb{R}^n$. These results were known for the convex case and played an essential role in the development of the theory. Because duality relations cannot be applied in the quasi-convex setting, new ingredients were introduced that give new understanding for the convex case as well.


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

  • [BBP] J. Bastero, J. Bernués and A. Peña, An extension of Milman's reverse Brunn-Minkowski inequality, GAFA 5 (1995), 572-581. MR 96g:26026
  • [BLM] J. Bourgain, J. Lindenstrauss, V. Milman, Approximation of zonoids by zonotopes. Acta Math. 162 (1989), no. 1-2, 73-141. MR 90g:46020
  • [BMMP] J. Bourgain, M. Meyer, V. Milman, A. Pajor, On a geometric inequality. Geometric aspects of functional analysis (1986/87), 271-282, Lecture Notes in Math., 1317, Springer, Berlin-New York, 1988. MR 89h:46012
  • [D] S.J. Dilworth, The dimension of Euclidean subspaces of quasi-normed spaces, Math. Proc. Camb. Phil. Soc., 97, 311-320, 1985. MR 86b:46003
  • [G] Y. Gordon, On Milman's inequality and random subspaces which escape through a mesh in $\mathbb{R}^n$. Geometric aspects of functional analysis (1986/87), 84-106, Lecture Notes in Math., 1317, Springer, Berlin-New York, 1988. MR 90b:46036
  • [GK] Y. Gordon, N.J. Kalton, Local structure theory for quasi-normed spaces, Bull. Sci. Math., 118, 441-453, 1994. MR 96c:46007
  • [JL] W.B. Johnson, J. Lindenstrauss, Extensions of Lipschitz mappings into a Hilbert space. Conference in modern analysis and probability (New Haven, Conn., 1982), 189-206. MR 86a:46018
  • [KPR] N.J. Kalton, N.T. Peck, J.W. Roberts, An $F$-space sampler, London Mathematical Society Lecture Note Series, 89, Cambridge University Press, Cambridge-New York, 1984. MR 87c:46002
  • [K] H. König, Eigenvalue Distribution of Compact Operators , Birkhäuser, 1986. MR 88j:47021
  • [LT] M. Ledoux, M. Talagrand, Probability in Banach spaces, Springer-Verlag, Berlin Heidelberg, 1991. MR 93c:60001
  • [M1] V.D. Milman, Almost Euclidean quotient spaces of subspaces of a finite-dimensional normed space. Proc. Amer. Math. Soc. 94 (1985), no. 3, 445-449. MR 87e:46026
  • [M2] V.D. Milman, Random subspaces of proportional dimension of finite-dimensional normed spaces: approach through the isoperimetric inequality. Banach spaces (Columbia, Mo., 1984), 106-115, Lecture Notes in Math., 1166, Springer, Berlin-New York, 1985. MR 87j:46037b
  • [M3] V.D. Milman, A note on a low $M\sp *$-estimate. Geometry of Banach spaces (Strobl, 1989), 219-229, London Math. Soc. Lecture Note Ser., 158, Cambridge Univ. Press, Cambridge, 1990. MR 92e:46018
  • [MP1] V.D. Milman, 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), 64-104, Lecture Notes in Math., 1376, Springer, Berlin-New York, 1989. MR 90g:52003
  • [MP2] V. Milman, A. Pajor, Cas limites dans des inégalités du type de Khinchine et applications géométriques. (French) [Limit cases of Khinchin-type inequalities and some geometric applications] C. R. Acad. Sci. Paris Sér. I Math. 308 (1989), no. 4, 91-96. MR 90d:52018
  • [MPi] V. Milman, G. Pisier, Banach spaces with a weak cotype $2$ property, Isr. J. Math., 54 (1980), 139-158. MR 88c:46022
  • [MS] V.D. Milman and G. Schechtman, Asymptotic theory of finite dimensional normed spaces , Lecture Notes in Math. 1200, Springer-Verlag (1986). MR 87m:46038
  • [PT] A. Pajor, N. Tomczak-Jaegermann, Subspaces of small codimension of finite-dimensional Banach spaces. Proc. Amer. Math. Soc. 97 (1986), no. 4, 637-642. MR 87i:46040
  • [P] G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge University Press (1989). MR 91d:52005
  • [R] S. Rolewicz, Metric linear spaces. Monografie Matematyczne, Tom. 56. [Mathematical Monographs, Vol. 56] PWN-Polish Scientific Publishers, Warsaw, 1972. MR 55:10993
  • [S] V.N. Sudakov, Gaussian measures, Cauchy measures and $\varepsilon $-entropy. Soviet. Math. Dokl., 10 (1969), 310-313.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 52C17, 46B07, 52A21, 52A30

Retrieve articles in all journals with MSC (1991): 52C17, 46B07, 52A21, 52A30


Additional Information

A. E. Litvak
Affiliation: Department of Mathematics, Tel Aviv University, Ramat Aviv, Israel
Address at time of publication: Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
Email: alexandr@math.tau.ac.il, alexandr@math.ualberta.ca

V. D. Milman
Email: vitali@math.tau.ac.il

A. Pajor
Affiliation: Universite de Marne-la-Valle, Equipe de Mathematiques, 2 rue de la Butte Verte, 93166, Noisy-le-Grand Cedex, France
Email: pajor@math.univ-mlv.fr

DOI: https://doi.org/10.1090/S0002-9939-99-04593-1
Received by editor(s): September 19, 1996
Received by editor(s) in revised form: June 14, 1997
Published electronically: January 29, 1999
Additional Notes: This research was done while the authors visited MSRI; we thank the Institute for its hospitality.
The first and second authors research was partially supported by BSF
Communicated by: Dale E. Alspach
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society