  We study effective potentials coming from compactifications of string theory.
We show that, under mild assumptions, such potentials are bounded from below in
four dimensions, giving an affirmative answer to a conjecture proposed by the
second author in arXiv:0911.3378v4 [hep-th]. We also derive some sufficient
conditions for the existence of critical points. All proofs and mathematical
hypotheses are discussed in the context of their relevance to the physics of
the problem.
