  We study a class of hypothesis testing problems in which, upon observing the
realization of an $n$-dimensional Gaussian vector, one has to decide whether
the vector was drawn from a standard normal distribution or, alternatively,
whether there is a subset of the components belonging to a certain given class
of sets whose elements have been ``contaminated,'' that is, have a mean
different from zero. We establish some general conditions under which testing
is possible and others under which testing is hopeless with a small risk. The
combinatorial and geometric structure of the class of sets is shown to play a
crucial role. The bounds are illustrated on various examples.
