By Ball K.

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**Example text**

Before we explain how this works, let’s look at our favourite examples. For specific norms it is usually much easier to compute the mean M by writing it as an integral with respect to Gaussian measure on Rn . As in Lecture 8 let µ be the standard Gaussian measure on Rn , with density 2 (2π)−n/2 e−|x| /2 . By using polar coordinates we can write S n−1 Γ(n/2) 1 θ dσ = √ x dµ(x) > √ x dµ(x). n Rn 2Γ((n + 1)/2) Rn The simplest norm for which to calculate is the 1 norm. Since the body we √ consider is supposed to have B2n as its maximal ellipsoid we must use nB1n , for which the corresponding norm is 1 x =√ n n |xi |.

The book of Pisier to which I have referred several times gives a more comprehensive account of many of the developments. I hope that readers of these notes may feel motivated to discover more. Acknowledgements I would like to thank Silvio Levy for his help in the preparation of these notes, and one of the workshop participants, John Mount, for proofreading the notes and suggesting several improvements. Finally, a very big thank you to my wife Sachiko Kusukawa for her great patience and constant love.

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.