On the isotropy constant of random convex sets
Author:
David AlonsoGutiérrez
Journal:
Proc. Amer. Math. Soc. 136 (2008), 32933300
MSC (2000):
Primary 52A20
Published electronically:
April 17, 2008
MathSciNet review:
2407095
Fulltext PDF Free Access
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Abstract: Let be the symmetric convex hull of independent random vectors uniformly distributed on the unit sphere of . We prove that, for every , the isotropy constant of is bounded by a constant with high probability, provided that .
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 Ball, K. Normed spaces with a weak GordonLewis property. Springer Lecture Notes in Math. 1470, Springer, Berlin (1991) pp. 3647. MR 1126735 (93e:46013)
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 Bourgain, J. On the distribution of polynomials on high dimensional convex sets, Springer Lecture Notes in Math. 1469, Springer, Berlin (1991), pp. 127137. MR 1122617 (92j:52007)
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 Bourgain, J.; Lindenstrauss, J.; Milman, V. D. Minkowski sums and symmetrizations. GAFA Seminar 198687, Springer Lecture Notes in Math. 1317, Springer, Berlin (1988), pp. 4466. MR 950975 (89g:46025)
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 Gluskin, E.D. Extremal properties of orthogonal parallelepipeds and their applications to the geometry of Banach spaces. Math. USSRSb. 64 (1989), pp. 8596. English translation. MR 945901 (89j:46016)
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 Klartag, B. On convex perturbations with a bounded isotropic constant. Geom. Funct. Anal. 16 (2006), no. 6, pp. 12741290. MR 2276540 (2007i:52005)
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 Klartag, B.; Kozma, G. On the hyperplane conjecture on random convex sets. (Preprint).
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 König, H.; Meyer, M.; Pajor, A. The isotropy constants of the Schatten classes are bounded. Math. Ann. 312 (1998), no. 4, pp. 773783. MR 1660231 (99j:52003)
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 Litvak, A.E.; Pajor, A.; Rudelson, M.; TomczakJaegermann, N. Smallest singular value of random matrices and geometry of random polytopes. Adv. Math. 195 (2005), pp. 491523. MR 2146352 (2006g:52009)
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Additional Information
David AlonsoGutiérrez
Affiliation:
Institute of Mathematics, Universidad de Zaragoza, 50009 Zaragoza, Spain
Email:
daalonso@unizar.es
DOI:
http://dx.doi.org/10.1090/S0002993908094872
PII:
S 00029939(08)094872
Received by editor(s):
July 10, 2007
Published electronically:
April 17, 2008
Additional Notes:
The author was supported by an FPU Scholarship from MEC (Spain), MCYT Grants (Spain) MTM200761446, DGA E64 and by Marie Curie RTN CT2004511953
Communicated by:
N. TomczakJaegermann
Article copyright:
© Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
