Deformations of unbounded convex bodies and hypersurfaces

We study the topology of the space $\partial{\mathcal K}^n$ of complete convex hypersurfaces of ${\bf R}^n$ which are homeomorphic to ${\bf R}^{n-1}$. In particular, using Minkowski sums, we construct a deformation retraction of $\partial{\mathcal K}^n$ onto the Grassmannian space of hyperplanes. So every hypersurface in $\partial{\mathcal K}^n$ 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 $\partial{\mathcal K}^n$ consisting of smooth, strictly convex, or positively curved hypersurfaces are each contractible, which settles a question of H. Rosenberg.