The statement that one region has more probability content than another with respect to all spherically symmetric (rotation invariant) distributions is a partial ordering among such regions. A simple geometric characterization of this ordering is given for star-shaped regions containing the origin. This characterization has several different interpretations. New techniques, inequalities, and examples are produced from this geometrical approach. The results have particular application to simultaneous confidence levels.
"Spherically Symmetric Probability Orderings Useful in Multiple Comparisons." Ann. Statist. 10 (4) 1253 - 1260, December, 1982. https://doi.org/10.1214/aos/1176345990