TITLE: What is known about the Gaussian measure of the unit ball in a Hilbert Space?
QUESTION [11 upvotes]: Let $X$ be an infinite dimensional separable Hilbert Space with norm $||\cdot||$ and let $\mu$ be a Gaussian measure on $X$ such that $\mu(X) = 1$.  What do we know about $\mu(B(0,1))$, where $B(0,1)$ is the unit ball w.r.t the norm?
This seems to me like a fundamental question but I cannot seem to find anything.  Any information/references would be most appreciated.
EDIT: A related question which is of interest to me:  Do there exist asymptotically tight bounds to 
$\int_{||u||> K}||u||^2 \mu(du)$?

REPLY [3 votes]: I think you can see the articles
entitled "concentration of measure phenomenon ". The idea is as
follows: Let $(X,d,\mu)$ be a metric measure space, such as
$\mu(X)=1$. Let
$$\alpha(\epsilon) = \sup {\mu(X \backslash A_\epsilon) \  | \ 
\mu(A) = 1/2 }$$ where
$$A_\epsilon = { x \  | \  d(x, A) < \epsilon }$$ is
the $\epsilon$-extension of a set $A$. The function $\alpha(.)$ is
called the concentration rate of the space $E$. The following
equivalent definition has many applications:$$\alpha(\epsilon) =
\sup { \mu( { F >= M + \epsilon }) },$$ where the supremum is
over all $1$-Lipschitz functions $F: X \to \mathbb{R}.$ For example
the median (or Levy mean) $M = \mathop{Med}(F) $ is defined by the
inequalities $$\mu ( F \geq M ) >= 1/2, \  \mu ( F <= M ) \geq
1/2.$$ More precisely, the space $X$ exhibits a concentration
phenomenon if $\alpha(\epsilon)$ decays very fast as $\epsilon$
grows. More formally, a family of metric measure spaces
$(X_n,d_n,\mu_n)$ is called a Levy family if the corresponding
concentration rates $\alpha(\epsilon)$ satisfy
$$\forall \epsilon > 0 \ \  \alpha_n(\epsilon) \to 0,$$ and a normal
Levy family if $$ \forall \epsilon \to 0 \ \  \alpha_n(\epsilon) =
O(\exp(-C n \epsilon^2))$$ for $C$ some positive constant. the last
inequality is obtained bay applying the "Hoeffding inequality" and in
the  case of  Hilbert space with concentration in small balls we do :
$$\forall (x_1,x_2)\in X^2, \ \  d(x_1,x_2)=\|x_1,x_2\|< r.$$