Let $(x_1,...,x_N)$ be points in $R^d$, and $\sigma=(\sigma_1,...,\sigma_N)$ are i.i.d. Rademacher variables (+1 or -1 with probability 0.5 each). Also let $z$ in some subset of $R^d$, and $\|\cdot\|_2$ will denote the Euclidean norm. I am looking for a good upper estimate of:

$E_{\sigma} \sup_z \| \sum_{n=1}^N \sigma_n \frac{x_n-z}{\|x_n-z\|_2} \|_2$.

So far I only got the upper bound of $N$ (by triangle inequality), and the lower bound $\sqrt{N}$ by Jensen inequality, and I would like to know if there is a way to improve on the upper estimate? Are there any conditions under which the upper bound would reduce to ${\mathcal O}(\sqrt{N})$?