Concentration of measure
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 measure space, . Let
is the -extension 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
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.
- Borell–TIS inequality
- Gaussian isoperimetric inequality
- McDiarmid's inequality
- Talagrand's concentration inequality
- Michel Talagrand, A New Look at Independence, The Annals of Probability, 1996, Vol. 24, No.1, 1-34
- "The concentration of , ubiquitous in the probability theory and statistical mechanics, was brought to geometry (starting from Banach spaces) by Vitali Milman, following the earlier work by Paul Lévy" - M. Gromov, Spaces and questions, GAFA 2000 (Tel Aviv, 1999), Geom. Funct. Anal. 2000, Special Volume, Part I, 118–161.
- "The idea of concentration of measure (which was discovered by V.Milman) is arguably one of the great ideas of analysis in our times. While its impact on Probability is only a small part of the whole picture, this impact should not be ignored." - M. Talagrand, A new look at independence, Ann. Probab. 24 (1996), no. 1, 1–34.
- Ledoux, Michel (2001). The Concentration of Measure Phenomenon. American Mathematical Society. ISBN 0-8218-2864-9.
- A. A. Giannopoulos and V. Milman, Concentration property on probability spaces, Advances in Mathematics 156 (2000), 77-106.