Summary: If \(X\sim P_ \theta\), \(\theta\in \Omega\), and \(\theta\sim G<<\mu\), where \(dG/d\mu\) belongs to the convex family \[ \Gamma_{L,U}=\{g: L\leq cg\leq U,\hbox{ for some }c>0\}, \] then the sets minimizing \(\lambda(S)\) subject to \(\inf_{G\in \Gamma_{L,U}}P_ G(S\mid X)\geq p\) are derived, where \(P_ G(S\mid X)\) is the posterior probability of \(S\) under the prior \(G\), and \(\lambda\) is any nonnegative measure on \(\Omega\) such that \(\mu<<\lambda<<\mu\).
Applications are shown to several multiparameter problems and connectedness (or disconnectedness) of these sets is considered. The problem of minimizing the diameter is also considered in a general probabilistic framework. It is proved that if \({\mathcal X}\) is any finite- dimensional Banach space with a convex norm, and \(\{P_ \alpha\}\) is a tight family of probability measures on the Borel \(\sigma\)-algebra of \({\mathcal X}\), then there always exists a closed connected set minimizing the diameter under the restriction \(\inf_{\alpha}P_ \alpha(S)\geq p\). It is also proved that if \(P\) is a spherical unimodal measure on \(\mathbb{R}^ m\), then volume ( Lebesgue measure) and diameter minimizing sets are the same.
A result of C. Borell [Periodica Math. Hungar. 6, 111-136 (1975; Zbl 0402.28007)] is then used to conclude that diameter minimizing sets are spheres whenever the underlying distribution \(P\) is symmetric, absolutey continuous and the density \(f\) is such that \(f^{-1/m}\) is convex. All standard symmetric multivariate densities satisfy this conditions. Applications are made to several Bayes and classical problems and admissibility implications of these results are discussed.