Concentration dimension
# Concentration dimension

In mathematics — specifically, in probability theory — the concentration dimension of a Banach space-valued random variable is a numerical measure of how "spread out" the random variable is compared to the norm on the space.

## Definition

Let (B, || ||) be a Banach space and let X be a Gaussian random variable taking values in B. That is, for every linear functional ℓ in the dual space $B^∗,$ the real-valued random variable ⟨ℓ, X⟩ has a normal distribution. Define

    $\sigma (X)=\sup \left\{\left.{\sqrt {\operatorname {E} [\langle \ell ,X\rangle ^{2}]}}\,\right|\,\ell \in B^{\ast },\|\ell \|\leq 1\right\}.$

Then the concentration dimension d(X) of X is defined by

    $d(X)={\frac {\operatorname {E} [\|X\|^{2}]}{\sigma (X)^{2}}}.$

## Examples

* If B is n-dimensional Euclidean space $R^n$ with its usual Euclidean norm, and X is a standard Gaussian random variable, then σ(X) = 1 and $E[||X||^2]$ = n, so d(X) = n.
* If B is $R^n$ with the supremum norm, then σ(X) = 1 but $E[||X||^2]$ (and hence d(X)) is of the order of log(n).
