Editing Concentration of measure (section)

==Concentration on the sphere==

The first example goes back to [[Paul Lévy (mathematician)|Paul Lévy]]. According to the [[spherical isoperimetric inequality]], among all subsets <math>A</math> of the sphere <math>S^n</math> with prescribed [[spherical measure]] <math>\sigma_n(A)</math>, the spherical cap
:<math> \left\{ x \in S^n | \mathrm{dist}(x, x_0) \leq R \right\}, </math>
for suitable <math>R</math>, has the smallest <math>\epsilon</math>-extension <math>A_\epsilon</math> (for any <math>\epsilon > 0</math>).

Applying this to sets of measure <math>\sigma_n(A) = 1/2</math> (where 
<math>\sigma_n(S^n) = 1</math>), one can deduce the following [[concentration inequality]]:
:<math>\sigma_n(A_\epsilon) \geq 1 - C \exp(- c n \epsilon^2) </math>,
where <math>C,c</math> are universal constants. Therefore <math>(S^n)_n</math> meet the definition above of a normal Lévy family.

[[Vitali Milman]] applied this fact to several problems in the local theory of Banach spaces, in particular, to give a new proof of [[Dvoretzky's theorem]].