  We define and study a new family of polytopes which are formed as convex
hulls of partial alternating sign matrices. We determine the inequality
descriptions, number of facets, and face lattices of these polytopes. We also
study partial permutohedra that we show arise naturally as projections of these
polytopes. We enumerate facets and also characterize the face lattices of
partial permutohedra in terms of chains in the Boolean lattice. Finally, we
have a result and a conjecture on the volume of partial permutohedra when one
parameter is fixed to be two.
