By Wolfgang Lück

ISBN-10: 3528032189

ISBN-13: 9783528032180

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We saw in Lecture 3 that in this situation K ⊂ nB2n , so x ≥ |x|/ n for all x, and hence 1 M≥√ . n But this estimate is useless, since it would not give slices of dimension bigger than 1. It is vital that we use the more detailed information provided by John’s Theorem. 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 .

M } in S k−1 is a δ-net for the sphere if every point of S k−1 is within AN ELEMENTARY INTRODUCTION TO MODERN CONVEX GEOMETRY 49 (Euclidean) distance δ of at least one ψi . The arguments in Lecture 2 show that S k−1 has a δ-net with no more than m= 4 δ k elements. The following lemma states that, indeed, pinning down the norm on a very fine net, pins it down everywhere. 2. Let · be a norm on some δ-net on S k−1 , we have Rk and suppose that for each point ψ of M (1 − γ) ≤ ψ ≤ M (1 + γ) for some γ > 0.

Notice that we certainly have M ≤ 1 since x ≤ |x| for all x. 3 we need to get a lower estimate for M of the order of √ log n √ . n AN ELEMENTARY INTRODUCTION TO MODERN CONVEX GEOMETRY 51 This is where we must use the fact that B2n is the maximal volume ellipsoid in √ √ K. We saw in Lecture 3 that in this situation K ⊂ nB2n , so x ≥ |x|/ n for all x, and hence 1 M≥√ . n But this estimate is useless, since it would not give slices of dimension bigger than 1. It is vital that we use the more detailed information provided by John’s Theorem.

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Algebraische Topologie by Wolfgang Lück


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