  Using a non-perturbative approximation for the partition function of a
complex scalar model, which features spontaneous symmetry breaking, we
explicitly derive the flattening of the effective potential in the region
limited by the minima of the bare potential. This flattening occurs in the
limit of infinite volume, and is a consequence of the summation over the
continuous set of saddle points which dominate the partition function. We also
prove the convexity of the effective potential and generalize the Maxwell
Construction for scalar theories with O(N) symmetry. Finally, we discuss why
the flattening of the effective potential cannot occur in the Abelian Higgs
theory.
