By Silvio Levy
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Funct. Anal. 1:2 (1991), 188–197. [Milman 1971] V. Milman, “A new proof of A. Dvoretzky’s theorem on cross-sections of convex bodies”, Funkcional. Anal. i Priloˇzen 5 (1971), 28–37. In Russian. [Milman 1985] V. Milman, “Almost Euclidean quotient spaces of subspaces of finite dimensional normed spaces”, Proc. Amer. Math. Soc. 94 (1985), 445–449. [Pisier 1989] G. Pisier, The volume of convex bodies and Banach space geometry, Tracts in Math. 94, Cambridge U. Press, Cambridge, 1989. [Rogers 1964] C.
The proof illustrates clearly why the Pr´ekopa–Leindler inequality makes things go smoothly. Although we only carried out the induction for sets, we required the one-dimensional result for the functions we get by scanning sets in Rn . To close this lecture we remark that the Brunn–Minkowski inequality has numerous extensions and variations, not only in convex geometry, but in combinatorics and information theory as well. One of the most surprising and delightful is a theorem of Busemann [1949].
T The method by which Bernstein’s inequality is proved has become an industry standard. Proof. We start by showing that, for each real λ, Eeλ Pa ε i i 2 ≤ eλ /2 . 1) The idea will then be that ai εi cannot be large too often, since, whenever it is large, its exponential is enormous. 1), we write Eeλ Pa ε n i i eλai εi =E 1 and use independence to deduce that this equals n Eeλai εi . 1 For each i the expectation is Eeλai εi = eλai + e−λai = cosh λai . 2 2 Now, cosh x ≤ ex /2 for any real x, so, for each i, 2 2 ai/2 Eeλai εi ≤ eλ Hence Eeλ Pa ε i i n ≤ 2 2 ai/2 eλ .