In general, a 1 — α confidence regionC(X)for a parameter θ ∈ Θ yields a test at level a forH: θ ∈ ΘHversusK: θ ∈ ΘCHwhenever we reject if C(X) ∪ ΘH= θ. We show under certain equivariance properties ofC(X)that for the case of convex alternatives, ΘCH, the level of the resulting test is in fact α/2. This extends recent findings for hyperrectangular alternatives as they occur in the multivariate bioequivalence problem. Furthermore, we apply the suggested test to ellipsoid-type alternatives instead of hyperrectangulars in the multivariate bioequivalence problem and to a problem occurring in neurophysiology. Finally, we compare our test numerically with existing methods.