# An Elementary Introduction to Modern Convex Geometry by Ball K.

By Ball K.

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For each positive ε, the Minkowski sum A + εB2n is exactly the set of points whose distance from A is at most ε. Let’s denote such an ε-neighbourhood Aε ; see Figure 25. The Brunn–Minkowski inequality shows that, if B is an Euclidean ball of the same volume as A, we have vol(Aε ) ≥ vol(Bε ) for any ε > 0. This formulation of the isoperimetric inequality makes much clearer the fact that it relates the measure and the metric on Rn . If we blow up a set in Rn using the metric, we increase the measure by at least as much as we would for a ball.

Concentration of Measure in Geometry The aim of this lecture is to describe geometric analogues of Bernstein’s deviation inequality. These geometric deviation estimates are closely related to isoperimetric inequalities. The phenomenon of which they form a part was introduced into the field by V. Milman: its development, especially by Milman himself, led to a new, probabilistic, understanding of the structure of convex bodies in high dimensions. The phenomenon was aptly named the concentration of measure.

A particularly elegant one [Gordon 1985] gives the estimate k ≥ cε2 log n (removing the logarithmic factor in ε−1 ), and this estimate is essentially best possible. We chose to describe Milman’s proof because it is conceptually easier to motivate and because the concentration of measure has many other uses. A few years ago, Schechtman found a way to eliminate the log factor within this approach, but we shall not introduce this subtlety here. We shall also not make any effort to be precise about the dependence upon ε.