Let $U$ denote the uniform distribution on the $n$-ball of radius $1$. What is the expected square-length of a vector under this distribution:

$$ \mathbf{E}_U[\|x\|^2] $$

By standard concentration-of-measure results almost all the probability measure is on vectors of length at least $(1-\nu)$ for any constant $\nu>0$, hence one can bound the above by $[1-\nu,1]$ for any constant $\nu$. However, is there a more precise statement?

More generally, let $U_{\epsilon}$ denote the uniform distribution on the set of vectors $x = (x_1,...,x_n)$ with $\|x\|\leq 1$ for which $x_1 \geq \epsilon$ ($\epsilon$ can depend on $n$). What is the expected square length of such vectors $x$ (i.e. conditioned on being at least $\epsilon$ away from the equator)

$$ f(\epsilon) := \mathbf{E}_{U_{\epsilon}}[\|x\|^2] $$

Note that here, concentration of measure does not apply in general because for fixed $\epsilon$ almost all vectors are located around the "equator", i.e. $|x_1| \leq \epsilon$, as $n \to \infty$.