arXiv math/06094472006-09-15math.DGmath.COmath.MG

Alexandrov's theorem, weighted Delaunay triangulations, and mixed volumes

Alexander I. Bobenko, Ivan Izmestiev

备注：: 53 pages, 16 figures
发表信息：: Annales de l'Institut Fourier 58:2 (2008) 447-505

英文摘要：

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.

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