Additional Applications of Levy Families in the Theory of Finite Dimensional Normed Spaces

Abstract

We saw in the previous chapters some applications of the concentration of measure phenomenon on Sⁿ. For example, we have proved in Chapters 4, 5 and 9 Dvoretzky’s Theorem and some of its generalizations. We have also seen applications of the same phenomenon and of some martingale techniques in the families {E ⁿⁿ₂ } ^∞_n=1 and {Π _n } ^∞_n=1 (see e.g., Theorem 9.7 and most of Chapter 8). In this chapter we will see a somewhat different kind of applications using some other Levy families. Some of the examples will be described only briefly leaving most of the details to the reader.