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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 .

**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 -dimensional space*, GAFA Seminar 1987-88, Springer Lecture Notes in Math.**1376**, Springer, Berlin (1989), pp. 64-104. MR**1008717 (90g:52003)**

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

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

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.