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