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In this paper we propose a general methodology, based on multiple testing,for testing that the mean of a Gaussian vector in R^n belongs to a convex set.We show that the test achieves its nominal level, and characterize a class ofvectors over which the tests achieve a prescribed power. In the functionalregression model this general methodology is applied to test some qualitativehypotheses on the regression function. For example, we test that the regressionfunction is positive, increasing, convex, or more generally, satisfies adifferential inequality. Uniform separation rates over classes of smoothfunctions are established and a comparison with other results in the literatureis provided. A simulation study evaluates some of the procedures for testingmonotonicity.

Testing convex hypotheses on the mean of a Gaussian vector. Application to testing qualitative hypotheses on a regression function