By Ball K.

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For the sake of readers who may not be familiar with probability theory, we also include a few words about independent random variables. To begin with, a probability measure µ on a set Ω is just a measure of total mass µ(Ω) = 1. Real-valued functions on Ω are called random variables and the integral of such a function X : Ω → R, its mean, is written EX and called the expectation of X. The variance of X is E(X − EX)2 . It is customary to suppress the reference to Ω when writing the measures of sets defined by random variables.

There are many other bodies and classes of bodies for which M can be efficiently estimated. For example, the correct order of the largest dimension of Euclidean slice of the np balls, was also computed in the paper [Figiel et al. 1977] mentioned earlier. 1) just as we do for the cube. The usual proof of this goes via the Dvoretzky–Rogers Lemma, which can be proved using John’s Theorem. This is done for example in [Pisier 1989]. Roughly speaking, the Dvoretzky–Rogers Lemma builds something like a cube around K, at least in a subspace of dimension about n2 , to which we then apply the result for the cube.

139 (1977), 53–94. [Garnaev and Gluskin 1984] A. Garnaev and E. Gluskin, “The widths of a Euclidean ball”, Dokl. A. N. USSR 277 (1984), 1048–1052. In Russian. [Gordon 1985] Y. Gordon, “Some inequalities for Gaussian processes and applications”, Israel J. Math. 50 (1985), 265–289. [Hoeffding 1963] W. Hoeffding, “Probability inequalities for sums of bounded random variables”, J. Amer. Statist. Assoc. 58 (1963), 13–30. [John 1948] F. John, “Extremum problems with inequalities as subsidiary conditions”, pp.

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Categories: Geometry And Topology