Measure Concentration and Almost Spherical Sections

Abstract

In the first two sections we are going to discuss measure concentration on a high-dimensional unit sphere. Roughly speaking, measure concentration says that if A ⊆ S ^{n -1} is a set occupying at least half of the sphere, then almost all points of S ^{n -1} are quite close to A, at distance about O(n ^-1/2). Measure concentration is an extremely useful technical tool in high-dimensional geometry. From the point of view of probability theory, it provides tail estimates for random variables defined on S ^{n -1}, and in this respect it resembles Chernoff-type tail estimates for the sums of independent random variables. But it is of a more general nature, more like tail estimates for Lipschitz functions on discrete spaces obtained using martingales.