  Let $\mathbf{X} = (X_i)_{1\leq i \leq n}$ be an i.i.d. sample of
square-integrable variables in $\mathbb{R}^d$, \GB{with common expectation
$\mu$ and covariance matrix $\Sigma$, both unknown.} We consider the problem of
testing if $\mu$ is $\eta$-close to zero, i.e. $\|\mu\| \leq \eta $ against
$\|\mu\| \geq (\eta + \delta)$; we also tackle the more general two-sample mean
closeness (also known as {\em relevant difference}) testing problem. The aim of
this paper is to obtain nonasymptotic upper and lower bounds on the minimal
separation distance $\delta$ such that we can control both the Type I and Type
II errors at a given level. The main technical tools are concentration
inequalities, first for a suitable estimator of $\|\mu\|^2$ used a test
statistic, and secondly for estimating the operator and Frobenius norms of
$\Sigma$ coming into the quantiles of said test statistic. These properties are
obtained for Gaussian and bounded distributions. A particular attention is
given to the dependence in the pseudo-dimension $d_*$ of the distribution,
defined as $d_* := \|\Sigma\|_2^2/\|\Sigma\|_\infty^2$. In particular, for
$\eta=0$, the minimum separation distance is ${\Theta}(
d_*^{\frac{1}{4}}\sqrt{\|\Sigma\|_\infty/n})$, in contrast with the minimax
estimation distance for $\mu$, which is
${\Theta}(d_e^{\frac{1}{2}}\sqrt{\|\Sigma\|_\infty/n})$ (where
$d_e:=\|\Sigma\|_1/\|\Sigma\|_\infty$). This generalizes a phenomenon spelled
out in particular by Baraud (2002).
