  We show that the size of a minimal simplicial cover of a polytope $P$ is a
lower bound for the size of a minimal triangulation of $P$, including ones with
extra vertices. We then use this fact to study minimal triangulations of cubes,
and we improve lower bounds for covers and triangulations in dimensions 4
through at least 12 (and possibly more dimensions as well). Important
ingredients are an analysis of the number of exterior faces that a simplex in
the cube can have of a specified dimension and volume, and a characterization
of corner simplices in terms of their exterior faces.
