We calculate one-loop corrections to four–point functions in the
scalar $\phi^4$ field theory in the spaces with constant curvature and
in flat space in the presence of a perfect mirror. We explain why the
calculations in Euclidian and Minkowskian signatures should not provide
the same result even at the leading order in non globaly hyperbolic
situations. Moreover, we either encounter non–local counter–terms or a
doubling of the beta–function. Our arguments are quite general and
applicable to other non-conformal theories in the same spaces.