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Euclidean Sections of Direct Sums of Normed Spaces

Published online by Cambridge University Press:  20 November 2018

A. E. Litvak  and
V. D. Milman
A. E. Litvak
Affiliation:
Department of Mathematics, Technion, Haifa, Israel, e-mail: alex@math.technion.ac.il Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, e-mail: alexandr@math.ualberta.ca
V. D. Milman
Affiliation:
Department of Mathematics, Tel Aviv University, Tel Aviv, Israel, e-mail: milman@post.tau.ac.il
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Abstract

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We study the dimension of “random” Euclidean sections of direct sums of normed spaces. We compare the obtained results with results from $[\text{LMS}]$, to show that for the direct sums the standard randomness with respect to the Haar measure on Grassmanian coincides with a much “weaker” randomness of “diagonal” subspaces (Corollary 1.4 and explanation after). We also add some relative information on “phase transition”.

Keywords

46B0746B0946B2052A21Dvoretzky theorem“random” Euclidean sectionphase transition in asymptotic convexity
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 46 , Issue 2 , 01 June 2003 , pp. 242 - 251
Copyright
Copyright © Canadian Mathematical Society 2003

References

[BLM] Bourgain, J., Lindenstrauss, J. and Milman, V. D., Approximation of zonoids by zonotopes. Acta Math. (1–2) 162 (1989), 73–141.Google Scholar
[CP] Carl, B. and Pajor, A., Gelfand numbers of operators with values in a Hilbert space. Invent.Math. (3) 94 (1988), 479–504.Google Scholar
[G1] Gluskin, E. D., Extremal properties of orthogonal parallelepipeds and their applications to the geometry of Banach spaces. Mat. Sb. (N.S.) (1) 136 (1988), 85–96; translation in Math. USSR-Sb. (1) 64 (1989), 85–96.Google Scholar
[G2] Gluskin, E. D., Deviation of a Gaussian vector from a subspace of , and random subspaces of . Algebra i Analiz (5) 1 (1989), 103–114; translation in Leningrad Math. J. (5) 1 (1990), 1165–1175.Google Scholar
[GGMP] Gordon, Y., Guédon, O., Meyer, M. and Pajor, A., On the Euclidean sections of some Banach spaces and operator spaces. Math. Scand., to appear.Google Scholar
[LMS] Litvak, A. E., Milman, V. D. and Schechtman, G., Averages of norms and quasi-norms. Math. Ann. 312 (1988), 95–124.Google Scholar
[MS1] Milman, V. D. and Schechtman, G., Asymptotic theory of finite-dimensional normed spaces. Lecture Notes in Math. 1200, Springer, Berlin, New York, 1985.Google Scholar
[MS2] Milman, V. D. and Schechtman, G., Global versus local asymptotic theories of finite dimensional normed spaces. Duke Math. J. (1) 90 (1997), 73–93.Google Scholar