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Euclidean Sections of Direct Sums of Normed Spaces

  • A. E. Litvak (a1) (a2) and V. D. Milman (a3)

Abstract

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”.

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References

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[BLM] Bourgain, J., Lindenstrauss, J. and Milman, V. D., Approximation of zonoids by zonotopes. Acta Math. (1–2) 162 (1989), 73141.
[CP] Carl, B. and Pajor, A., Gelfand numbers of operators with values in a Hilbert space. Invent.Math. (3) 94 (1988), 479504.
[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), 8596; translation in Math. USSR-Sb. (1) 64 (1989), 8596.
[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), 11651175.
[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.
[LMS] Litvak, A. E., Milman, V. D. and Schechtman, G., Averages of norms and quasi-norms. Math. Ann. 312 (1988), 95124.
[MS1] Milman, V. D. and Schechtman, G., Asymptotic theory of finite-dimensional normed spaces. Lecture Notes in Math. 1200, Springer, Berlin, New York, 1985.
[MS2] Milman, V. D. and Schechtman, G., Global versus local asymptotic theories of finite dimensional normed spaces. Duke Math. J. (1) 90 (1997), 7393.
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Euclidean Sections of Direct Sums of Normed Spaces

  • A. E. Litvak (a1) (a2) and V. D. Milman (a3)

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