The paper studies hypothesis testing problems for the mean of a vector variate having a multivariate normal distribution, in cases where the alternative is restricted by a number of linear inequalities. A new criterion which can be regarded as a generalization of the "maximin-$r^2$" criterion of Abelson and Tukey (cf. [1]) is introduced: we try to obtain tests which are "most stringent" among the "somewhere most powerful" tests, (Section 2). For an important class of testing problems (Section 3) such tests can be characterized by a (half-) line $l_0$ minimizing a maximum angle (Sections 4, 6 and 7). This (half-) line $l_0$ can be obtained by means of a method described by Abelson and Tukey (Section 5). The theory of this paper can be applied to a large number of actual testing problems. Such applications, (one of which is treated in Section 8), generalizations and details of the theory of this paper will be considered in the thesis [5].