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On the isotropy constant of random convex sets


Author: David Alonso-Gutiérrez
Journal: Proc. Amer. Math. Soc. 136 (2008), 3293-3300
MSC (2000): Primary 52A20
DOI: https://doi.org/10.1090/S0002-9939-08-09487-2
Published electronically: April 17, 2008
MathSciNet review: 2407095
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Abstract: Let $ K$ be the symmetric convex hull of $ m$ independent random vectors uniformly distributed on the unit sphere of $ \mathbb{R}^n$. We prove that, for every $ \delta>0$, the isotropy constant of $ K$ is bounded by a constant $ c(\delta)$ with high probability, provided that $ m\geq (1+\delta)n$.


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  • 1. Ball, K. Normed spaces with a weak Gordon-Lewis property. Springer Lecture Notes in Math. 1470, Springer, Berlin (1991) pp. 36-47. MR 1126735 (93e:46013)
  • 2. Bourgain, J. On the distribution of polynomials on high dimensional convex sets, Springer Lecture Notes in Math. 1469, Springer, Berlin (1991), pp. 127-137. MR 1122617 (92j:52007)
  • 3. Bourgain, J.; Lindenstrauss, J.; Milman, V. D. Minkowski sums and symmetrizations. GAFA Seminar 1986-87, Springer Lecture Notes in Math. 1317, Springer, Berlin (1988), pp. 44-66. MR 950975 (89g:46025)
  • 4. Gluskin, E.D. Extremal properties of orthogonal parallelepipeds and their applications to the geometry of Banach spaces. Math. USSR-Sb. 64 (1989), pp. 85-96. English translation. MR 945901 (89j:46016)
  • 5. Klartag, B. On convex perturbations with a bounded isotropic constant. Geom. Funct. Anal. 16 (2006), no. 6, pp. 1274-1290. MR 2276540 (2007i:52005)
  • 6. Klartag, B.; Kozma, G. On the hyperplane conjecture on random convex sets. (Preprint).
  • 7. König, H.; Meyer, M.; Pajor, A. The isotropy constants of the Schatten classes are bounded. Math. Ann. 312 (1998), no. 4, pp. 773-783. MR 1660231 (99j:52003)
  • 8. Litvak, A.E.; Pajor, A.; Rudelson, M.; Tomczak-Jaegermann, N. Smallest singular value of random matrices and geometry of random polytopes. Adv. Math. 195 (2005), pp. 491-523. MR 2146352 (2006g:52009)
  • 9. Milman, V.; Pajor, A. Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed $ n$-dimensional space, GAFA Seminar 1987-88, Springer Lecture Notes in Math. 1376, Springer, Berlin (1989), pp. 64-104. MR 1008717 (90g:52003)

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Additional Information

David Alonso-Gutiérrez
Affiliation: Institute of Mathematics, Universidad de Zaragoza, 50009 Zaragoza, Spain
Email: daalonso@unizar.es

DOI: https://doi.org/10.1090/S0002-9939-08-09487-2
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) MTM2007-61446, DGA E-64 and by Marie Curie RTN CT-2004-511953
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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