In mathematics, concentration of measure (about a median) is a principle that is applied in measure theory, probability and combinatorics, and has consequences for other fields such as Banach space theory. Informally, it states that "A random variable that depends in a Lipschitz way on many independent variables (but not too much on any of them) is essentially constant". The concentration of measure phenomenon was put forth in the early 1970s by Vitali Milman in his works on the local theory of Banach spaces, extending an idea going back to the work of Paul Lévy. It was further developed in the works of Milman and Gromov, Maurey, Pisier, Schechtman, Talagrand, Ledoux, and others. The general setting Let be a metric space with a measure on the Borel sets with . Let where is the -extension (also called -fattening in the context of the Hausdorff distance) of a set . The function is called the concentration rate of the space . The following equivalent definition has many applications: where the supremum is over all 1-Lipschitz functions , and the median (or Levy mean) is defined by the inequalities Informally, the space exhibits a concentration phenomenon if decays very fast as grows. More formally, a family of metric measure spaces is called a Lévy family if the corresponding concentration rates satisfy and a normal Lévy family if for some constants . For examples see below. Concentration on the sphere The first example goes back to Paul Lévy. According to the spherical isoperimetric inequality, among all subsets of the sphere with prescribed spherical measure , the spherical cap for suitable , has the smallest -extension (for any ). Applying this to sets of measure (where ), one can deduce the following concentration inequality: , where are universal constants. Therefore 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. Concentration of measure in physics All classical statistical physics is based on the concentration of measure phenomena: The fundamental idea (‘theorem’) about equivalence of ensembles in thermodynamic limit (Gibbs, 1902 and Einstein, 1902-1904) is exactly the thin shell concentration theorem. For each mechanical system consider the phase space equipped by the invariant Liouville measure (the phase volume) and conserving energy E. The microcanonical ensemble is just an invariant distribution over