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By Vitali D. Milman

This ebook offers with the geometrical constitution of finite dimensional normed areas, because the size grows to infinity. this can be a a part of what got here to be referred to as the neighborhood concept of Banach areas (this identify was once derived from the truth that in its first phases, this concept dealt frequently with pertaining to the constitution of endless dimensional Banach areas to the constitution in their lattice of finite dimensional subspaces). Our function during this booklet is to introduce the reader to a couple of the consequences, difficulties, and as a rule tools constructed within the neighborhood concept, within the previous few years. This under no circumstances is an entire survey of this vast zone. a number of the major issues we don't speak about listed here are pointed out within the Notes and feedback part. numerous books seemed lately or are going to seem presently, which hide a lot of the fabric now not lined during this e-book. between those are Pisier's [Pis6] the place factorization theorems with regards to Grothendieck's theorem are commonly mentioned, and Tomczak-Jaegermann's [T-Jl] the place operator beliefs and distances among finite dimensional normed areas are studied intimately. one other similar booklet is Pietch's [Pie].

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C) j=l n n :S E exp(>. 2. L j=l Putting>. c). 2Lj=1I1djll~, we get n P(/ - EI ~ c) :S exp(-c 2/4 L IIdjll~)· j=l Similarly, n P(EI - I ~ c) :S exp(-c 2/4 L Ildjll~) j=l and P(II - Ell ~ c) :S P(/ - EI ~ c) + P(EI - I n ~ c) :S 2exp( _c 2 /4 L IIdjll~). 5. 4. THEOREM: The family TIn of permutation of {I, ... n} with the metric d(ll", e) = ~1{ijll"(i) t= e(i)} I and the uniform measure P (assigning mass tion) is a normal Levy family with constants C1 = 2 and C2 = 1/64. 6. PROOF: Let 1j, 0 ::; j ::; n, be the algebra of subsets of TIn generated by the atoms {Ai, ,..

Appendix I, written by Gromov, contains the proof together with the necessary definitions. lx be the normalized riemannian volume element on a connected riemannian manifold without boundary X and let R(X) be the Ricci curvature of X. l being the normalized Haar measure on T· sn. 2. leads to the following examples. 1. Cl = V1T/8,C2 = 1/8. 2. Similarly for each m the family X n = sn the product measure and the metric X sn X ••• Sn(m - times), n = 1,2, ... 6. Cl = V1T/8, C2 = 1/2. Next we show that homogeneous spaces of SOn inherit the property of being Levy family.

8. Given a measure space (11, Y,p,) and a Banach space X, we define the space L 2 (X) = L 2 (X,I1) as the space of all measurable functions from (11, Y) to X with IIIII = 1/2 ( < 111/(w)llidp,) 00 • In the cases we shall be interested in, the space 11 will be finite and no problem of measurability will arise. Actually, the only measure space relevant to us is the space {-I, 1}n with p, - the normalized counting measure. For IE L 2(X), 9 E L2(X') we define < g,1 >= J g(w)(f(w))dp,. relation. 9. 11'lli,(x) = II·IIL,(X·), when X = This defines a duality L 2 (X') with equality of is finite dimensional.

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