Minimaxity under the half-Cauchy prior

This is a follow-up paper of Polson and Scott (2012, Bayesian Analysis), which claimed that the half-Cauchy prior is a sensible default prior for a scale parameter in hierarchical models. For estimation of a $p$-variate normal mean under the quadratic loss, they demonstrated that the Bayes estimator with respect to the half-Cauchy prior seems to be minimax through numerical experiments. In this paper, we theoretically establish the minimaxity of the corresponding Bayes estimator using the interval arithmetic.