Deformations of unbounded convex bodies and hypersurfaces

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We study the topology of the space ⁿ of complete convex hypersurfaces of ⁿ which are homeomorphic to ⁿ⁻¹. In particular, using Minkowski sums, we construct a deformation retraction of ⁿ onto the Grassmannian space of hyperplanes. So every hypersurface in 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 ⁿ consisting of smooth, strictly convex, or positively curved hypersurfaces are each contractible, which settles a question of H. Rosenberg.