  We obtain effective potential of $O(N)$-symmetric $\phi^4$ theory for large
$N$ starting with a finite lattice system and taking the thermodynamic limit
with great care. In the thermodynamic limit, it is globally real-valued and
convex in both the symmetric and the broken phases. In particular, it has a
flat bottom in the broken phase. Taking the continuum limit, we discuss
renormalization effects to the flat bottom and exhibit the effective potential
of the continuum theory in three and four dimensions.On the other hand the
effective potential is nonconvex in a finite lattice system. Our numerical
study shows that the barrier height of the effective potential flattens as a
linear size of the system becomes large. It decreases obeying power law and the
exponent is about $-2$. The result is clearly understood from dominance of
configurations with slowly-rotating field in one direction.
