  The Ekeland variational principle implies what can be regarded as a strong
version, in the $C^1$ category, of the Yau minimum principle: under the
appropriate hypotheses {\it every} minimizing sequence admits a {\it good
shadow}, a second minimizing sequence that has good properties and is
asymptotic to the original one. Using arguments from dynamical systems, we give
another proof of this result and also establish, with the aid of Gromov's
theorem on monotonicity of volume ratios, a special case of a conjecture
claiming the existence of good shadows in the original $C^2$ setting of the Yau
minimum principle. The interest in having an abundance of good shadows stems
from the fact that this is a desirable property if one wants to refine the
applications of the asymptotic minimum principle, as it allows for information
to be localized at infinity. These ideas are applied in this paper to the study
of the convex hulls of complete submanifolds of Euclidean $n$-space that have
controlled Grassmanian-valued Gauss maps.
