Bayes estimation of the mean of a multivariate normal distribution is considered under quadratic loss. We show that, when particular spherical priors are used, the superharmonicity of the square root of the marginal density provides a viable method for constructing (possibly proper) Bayes (and admissible) minimax estimators. Examples illustrate the theory; most notably it is shown that a multivariate Student-$t$ prior yields a proper Bayes minimax estimate.
"On the construction of Bayes minimax estimators." Ann. Statist. 26 (2) 660 - 671, April 1998. https://doi.org/10.1214/aos/1028144853