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We study the topology of the space $\d\K^n$ of complete convex hypersurfacesof $\R^n$ which are homeomorphic to $\R^{n-1}$. In particular, using Minkowskisums, we construct a deformation retraction of $\d\K^n$ onto the Grassmannianspace of hyperplanes. So every hypersurface in $\d \K^n$ may be flattened in acanonical way. Further, the total curvature of each hypersurface evolvescontinuously 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.

Deformations of unbounded convex bodies and hypersurfaces