By Ball K.
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22 (1976), 366–389. [Brøndsted 1983] A. Brøndsted, An introduction to convex polytopes, Graduate Texts in Math. 90, Springer, New York, 1983. 54 KEITH BALL [Busemann 1949] H. Busemann, “A theorem on convex bodies of the Brunn–Minkowski type”, Proc. Nat. Acad. Sci. USA 35 (1949), 27–31. [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.
Xn ) is distributed according to Lebesgue measure on n a2i = 1, then ai Xi is the distance of the point the cube − 12 , 12 ⊂ Rn . If n (X1 , X2 , . . , Xn ) from the subspace of R orthogonal to (a1 , a2 , . . , an ). 2) says that most of the mass of the cube lies close to any subspace of Rn , which is reminiscent of the situation for the Euclidean ball described in Lecture 1. Lecture 8. Concentration of Measure in Geometry The aim of this lecture is to describe geometric analogues of Bernstein’s deviation inequality.
The phenomenon just described becomes even more striking when reinterpreted in terms of Lipschitz functions. Suppose f : S n−1 → R is a function on the sphere that is 1-Lipschitz: that is, for any pair of points θ and φ on the sphere, |f (θ) − f (φ)| ≤ |θ − φ| . There is at least one number M , the median of f , for which both the sets (f ≤ M ) and (f ≥ M ) have measure at least 12 . If a point x has distance at most ε from ε Figure 27. An ε-neighbourhood of a hemisphere. 44 KEITH BALL (f ≤ M ), then (since f is 1-Lipschitz) f (x) ≤ M + ε.
An Elementary Introduction to Modern Convex Geometry by Ball K.