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 $μ$ and covariance matrix $Σ$, both unknown.} We consider the problem of testing if $μ$ is $η$-close to zero, i.e. $\|μ\| \leq η$ against $\|μ\| \geq (η+ δ)$; 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 $δ$ 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 $\|μ\|^2$ used a test statistic, and secondly for estimating the operator and Frobenius norms of $Σ$ 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 $d_* := \|Σ\|_2^2/\|Σ\|_\infty^2$. In particular, for $η=0$, the minimum separation distance is $Θ( d_*^{\frac{1}{4}}\sqrt{\|Σ\|_\infty/n})$, in contrast with the minimax estimation distance for $μ$, which is $Θ(d_e^{\frac{1}{2}}\sqrt{\|Σ\|_\infty/n})$ (where $d_e:=\|Σ\|_1/\|Σ\|_\infty$). This generalizes a phenomenon spelled out in particular by Baraud (2002).