  We present a constructive proof of Alexandrov's theorem regarding the
existence of a convex polytope with a given metric on the boundary. The
polytope is obtained as a result of a certain deformation in the class of
generalized convex polytopes with the given boundary. We study the space of
generalized convex polytopes and discover a relation with the weighted Delaunay
triangulations of polyhedral surfaces. The existence of the deformation follows
from the non-degeneracy of the Hessian of the total scalar curvature of a
positively curved generalized convex polytope. The latter is shown to be equal
to the Hessian of the volume of the dual generalized polyhedron. We prove the
non-degeneracy by generalizing the Alexandrov-Fenchel inequality. Our
construction of a convex polytope from a given metric is implemented in a
computer program.
