  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 of
vectors over which the tests achieve a prescribed power. In the functional
regression model this general methodology is applied to test some qualitative
hypotheses on the regression function. For example, we test that the regression
function is positive, increasing, convex, or more generally, satisfies a
differential inequality. Uniform separation rates over classes of smooth
functions are established and a comparison with other results in the literature
is provided. A simulation study evaluates some of the procedures for testing
monotonicity.
