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Euclidean sections of convex bodies, series of lectures

This is a somewhat expanded form of a four hours course given, with small variations, first at the educational workshop Probabilistic methods in Geometry, Bedlewo, Poland, July 6-12, 2008 and a few weeks later at the Summer school on Fourier analytic and probabilistic methods in geometric functional analysis and convexity, Kent, Ohio, August 13-20, 2008. The main part of these notes gives yet another exposition of Dvoretzky's theorem on Euclidean sections of convex bodies with a proof based on Milman's. This material is by now quite standard. Towards the end of these notes we discuss issues related to fine estimates in Dvoretzky's theorem and there there are some results that didn't appear in print before. In particular there is an exposition of an unpublished result of Figiel (Claim \ref{claim:figiel}) which gives an upper bound on the possible dependence on $\e$ in Milman's theorem. We would like to thank Tadek Figiel for allowing us to include it here. There is also a better version of the proof of one of the results from \cite{sc2} giving a lower bound on the dependence on $\e$ in Dvoretzky's theorem. The improvement is in the statement and proof of Proposition \ref{prop:main} here which is a stronger version of the corresponding Corollary 1 in \cite{sc2}.