  We define the alternating sign matrix polytope as the convex hull of nxn
alternating sign matrices and prove its equivalent description in terms of
inequalities. This is analogous to the well known result of Birkhoff and von
Neumann that the convex hull of the permutation matrices equals the set of all
nonnegative doubly stochastic matrices. We count the facets and vertices of the
alternating sign matrix polytope and describe its projection to the
permutohedron as well as give a complete characterization of its face lattice
in terms of modified square ice configurations. Furthermore we prove that the
dimension of any face can be easily determined from this characterization.
