Georgia Institute of Technology College of Sciences

We study the topology of the space bd 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 bd K^n onto the Grassmannian space of hyperplanes. So every hypersurface in bd 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 bd K^n consisting of smooth, strictly convex, or positively curved hypersurfaces are each contractible, which settles a question of H. Rosenberg.