  We study the topology of the space $\d\K^n$ of complete convex hypersurfaces
of $\R^n$ which are homeomorphic to $\R^{n-1}$. In particular, using Minkowski
sums, we construct a deformation retraction of $\d\K^n$ onto the Grassmannian
space of hyperplanes. So every hypersurface in $\d \K^n$ may be flattened in a
canonical way. Further, the total curvature of each hypersurface evolves
continuously and monotonically under this deformation. We also show that,
modulo proper rotations, the subspaces of $\d\K^n$ consisting of smooth,
strictly convex, or positively curved hypersurfaces are each contractible,
which settles a question of H. Rosenberg.
