  Let $(X\_1,\ldots,X\_n)$ be a $d$-dimensional i.i.d sample from a
distribution with density $f$. The problem of detection of a two-component
mixture is considered. Our aim is to decide whether $f$ is the density of a
standard Gaussian random $d$-vector ($f=\phi\_d$) against $f$ is a
two-component mixture: $f=(1-\varepsilon)\phi\_d +\varepsilon \phi\_d (.-\mu)$
where $(\varepsilon,\mu)$ are unknown parameters. Optimal separation conditions
on $\varepsilon, \mu, n$ and the dimension $d$ are established, allowing to
separate both hypotheses with prescribed errors. Several testing procedures are
proposed and two alternative subsets are considered.
