In the first nontrivial case, dimension $p = 2$ and sample size $N = 3$, it is proved that Hotelling's $T^2$ test of level $\alpha$ maximizes, among all level $\alpha$ tests, the minimum power on each of the usual contours where the $T^2$ test has constant power. A corollary is that the $T^2$ test is most stringent of level $\alpha$ in this case.
"Minimax Character of Hotelling's $T^2$ Test in the Simplest Case." Ann. Math. Statist. 34 (4) 1524 - 1535, December, 1963. https://doi.org/10.1214/aoms/1177703884