  What is the minimal closed cone containing all $f$-vectors of cubical
$d$-polytopes? We construct cubical polytopes showing that this cone, expressed
in the cubical $g$-vector coordinates, contains the nonnegative $g$-orthant,
thus verifying one direction of the Cubical Generalized Lower Bound Conjecture
of Babson, Billera and Chan. Our polytopes also show that a natural cubical
analogue of the simplicial Generalized Lower Bound Theorem does not hold.
