An Elementary Introduction to Modern Convex Geometry by Ball K.

By Ball K.

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In Russian. [Gordon 1985] Y. Gordon, “Some inequalities for Gaussian processes and applications”, Israel J. Math. 50 (1985), 265–289. [Hoeffding 1963] W. Hoeffding, “Probability inequalities for sums of bounded random variables”, J. Amer. Statist. Assoc. 58 (1963), 13–30. [John 1948] F. John, “Extremum problems with inequalities as subsidiary conditions”, pp. 187–204 in Studies and essays presented to R. Courant on his 60th birthday (Jan. 8, 1948), Interscience, New York, 1948. [Johnson and Schechtman 1982] W.

Lieb, “Best constants in Young’s inequality, its converse and its generalization to more than three functions”, Advances in Math. 20 (1976), 151–173. [Brascamp and Lieb 1976b] H. J. Brascamp and E. H. Lieb, “On extensions of the Brunn–Minkowski and Pr´ekopa–Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation”, J. Funct. Anal. 22 (1976), 366–389. [Brøndsted 1983] A. Brøndsted, An introduction to convex polytopes, Graduate Texts in Math.

We shall deduce the first assertion directly from the Pr´ekopa–Leindler inequality (with λ = 12 ) of Lecture 5. To this end, define functions f , g, and m on Rn , as follows: 2 f (x) = ed(x,A) /4 γ(x), g(x) = χA (x) γ(x), m(x) = γ(x), where γ is the Gaussian density. The assertion to be proved is that 2 ed(x,A) /4 dµ µ(A) ≤ 1, which translates directly into the inequality 2 Rn f Rn g ≤ Rn m . By the Pr´ekopa–Leindler inequality it is enough to check that, for any x and y in Rn , f (x)g(y) ≤ m x+y 2 2 .

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