In Section 3 we shall prove a theorem based on a method of A. Birnbaum [1] and E. Lehmann concerning the admissibility of certain tests of simple hypotheses in multivariate exponential families. In Section 4 we compute the supporting hyperplanes of the convex acceptance region in some of the most common applications of Hotelling's $T^2$-test and show that the theorem of Section 3 implies the admissibility of this test. In Section 5 we point out some of the limitations of the method of this paper.