  We consider geometrical optimization problems related to optimizing the error
probability in the presence of a Gaussian noise. One famous questions in the
field is the "weak simplex conjecture". We discuss possible approaches to it,
and state related conjectures about the Gaussian measure, in particular, the
conjecture about minimizing of the Gaussian measure of a simplex. We also
consider antipodal codes, apply the \v{S}id\'ak inequality and establish some
theoretical and some numerical results about their optimality.
