This paper contains details of the results announced in the abstract by the authors (1962). Techniques are developed for proving local minimax and "type $D$" properties and asymptotic (that is, far in distance from the null hypothesis) minimax properties in complex testing problems where exact minimax results seem difficult to obtain. The techniques are illustrated in the settings where Hotelling's $T^2$ test and the test based on the squared sample multiple correlation coefficient $R^2$ are customarily employed.
"Local and Asymptotic Minimax Properties of Multivariate Tests." Ann. Math. Statist. 35 (1) 21 - 35, March, 1964. https://doi.org/10.1214/aoms/1177703730