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

Alexander I. Bobenko·Ivan Izmestiev
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Abstract

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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Alexander I. Bobenko, Ivan Izmestiev. 2006-09-15. Alexandrov's theorem, weighted Delaunay triangulations, and mixed volumes. https://arxiv.org/abs/math/0609447

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