On detection of Gaussian stochastic sequences

We consider the minimax detection problem for a Gaussian random signal vector in white Gaussian additive noise. It is assumed that an unknown vector $\boldsymbol\sigma$ of signal vector intensities belongs to a given set $\mathcal E$. We investigate when it is possible to replace the set $\mathcal E$ with a smaller set $\mathcal E_0$ without loss of quality (and, in particular, replace it with a single point  $\boldsymbol\sigma_0$).