Sudeep kamath concentration of measure 1
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".
Contents
- Sudeep kamath concentration of measure 1
- Lecture 18 concentration of measure
- The general setting
- Concentration on the sphere
- Other examples
- References
The c.o.m. 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, Shechtman, Talagrand, Ledoux, and others.
Lecture 18 concentration of measure
The general setting
Let
where
is the
The function
where the supremum is over all 1-Lipschitz functions
Informally, the space
and a normal Lévy family if
for some constants
Concentration on the sphere
The first example goes back to Paul Lévy. According to the spherical isoperimetric inequality, among all subsets
for suitable
Applying this to sets of measure
where
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.