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2 Suppose t ≥ 0. Whenever ai εi ≥ t, we will have et ai εi ≥ et . 1). So Prob ai ε i ≥ t 2 ≤ e−t /2 , and in a similar way we get Prob ai εi ≤ −t 2 ≤ e−t /2 . Putting these inequalities together we get Prob ai ε i ≥ t 2 ≤ 2e−t /2 .  In the next lecture we shall see that deviation estimates that look like Bernstein’s inequality hold for a wide class of functions in several geometric settings. 2) holds for a2i = 1, if the ±1 valued random variables εi are replaced by independent random variables Xi each of which is uniformly distributed on the interval − 21 , 12 .

4 (The Pr´ ekopa–Leindler inequality). If f , g and m are nonnegative measurable functions on Rn , λ ∈ (0, 1) and for all x and y in Rn , m ((1 − λ)x + λy) ≥ f (x)1−λ g(y)λ then 1−λ m≥ λ . 2) It is perhaps helpful to notice that the Pr´ekopa–Leindler inequality looks like H¨ older’s inequality, backwards. If f and g were given and we set m(z) = f (z)1−λ g(z)λ (for each z), then H¨ older’s inequality says that 1−λ m≤ f λ g . ) The difference between Pr´ekopa–Leindler and H¨ older is that, in the former, the value m(z) may be much larger since it is a supremum over many pairs (x, y) satisfying z = (1 − λ)x + λy rather than just the pair (z, z).

M. Ball, “Volume ratios and a reverse isoperimetric inequality”, J. London Math. Soc. 44 (1991), 351–359. [Beckner 1975] W. Beckner, “Inequalities in Fourier analysis”, Ann. of Math. 102 (1975), 159–182. [Borell 1975] C. Borell, “The Brunn–Minkowski inequality in Gauss space”, Inventiones Math. 30 (1975), 205–216. [Bourgain and Milman 1987] J. Bourgain and V. Milman, “New volume ratio properties for convex symmetric bodies in Rn ”, Invent. Math. 88 (1987), 319–340. [Bourgain et al. 1989] J. Bourgain, J.

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An Elementary Introduction to Modern Convex Geometry by Ball K.

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