 Minimaxity of the usual multivariate normal mean estimator

For $X\sim N_p(\theta ,I_p)$ under quadratic loss, the estimator ${\delta }_0(X)=X$ has constant risk $p$ and is minimax. Gaussian priors $N_p(0,c^2{I}_p)$ have Bayes risks $p{c}^2/(1+c^2)$, which tend to $p$ and therefore give the matching minimax lower bound.