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. 1. Introduction. When estimating the mean of a multivariate distribution, the two dominant approaches are the minimax approach and variants of the Bayes paradigm. The first has received the most extensive development while the second is most used in practice, due to its great flexibility. See [4] for a study of the interface between these two approaches. The problem of both methods is that neither necessarily leads to admissible estimators (hierarchical Bayes estimators are often only generalized Bayes estimators). Even if admissibility may provide nonreasonable estimators, it is a criterion which can be desirable when combining minimaxity and Bayesian properties. Indeed, since the sampling distribution is normal, under quadratic loss, the Bayes estimator is unique provided the Bayes risk is finite, so that the proper Bayes estimator is admissible (see [16]). In [5], Brown conjectured that, for estimating a multivariate normal mean using quadratic loss, a proper Bayes minimax estimator does not exist for four or less dimensions. This conjecture was proved by Strawderman [21], who also settled the conjecture for dimensions five or more that such estimators do indeed exist. Stein [19] obtains minimaxity of a general estimator δ of the form δ� x �= ∞