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Extra resources for An Elementary Introduction to Modern Convex Geometry

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1) is smallest. This is proved by showing that for each r, the Gaussian measure of rK is at most that of [−r, r]n . The details are left to the reader. This last lecture has described work that dates back to the seventies. Although some of the material in earlier lectures is more recent (and some is much older), I have really only scratched the surface of what has been done in the last twenty years. The book of Pisier to which I have referred several times gives a more comprehensive account of many of the developments.

Figiel et al. 1977] T. Figiel, J. Lindenstrauss, and V. Milman, “The dimension of almost spherical sections of convex bodies”, Acta Math. 139 (1977), 53–94. [Garnaev and Gluskin 1984] A. Garnaev and E. Gluskin, “The widths of a Euclidean ball”, Dokl. A. N. USSR 277 (1984), 1048–1052. 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.

To answer this question we need to rewrite Young’s inequality slightly. If 1/r +1/s = 1, the Ls norm f ∗g s can be realised as R for some function h with h 2, then r (f ∗ g)(x)h(x) = 1. So the inequality says that, if 1/p+1/q+1/r = f (y)g(x − y)h(x) dy dx ≤ f p g q h r . AN ELEMENTARY INTRODUCTION TO MODERN CONVEX GEOMETRY 35 We may rewrite the inequality again with h(−x) in place of h(x), since this doesn’t affect h r : f (y)g(x − y)h(−x) dy dx ≤ f p g q h r . 1) This can be written in a more symmetric form via the map from R2 into R3 that takes (x, y) to (y, x−y, −x) =: (u, v, w).

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