Alexandrov’s theorem, weighted Delaunay triangulations, and mixed volumes
Annales de l'Institut Fourier, Volume 58 (2008) no. 2, p. 447-505

We present a constructive proof of Alexandrov’s theorem on the existence of a convex polytope with a given metric on the boundary. The polytope is obtained by deforming certain generalized convex polytopes with the given boundary. We study the space of generalized convex polytopes and discover a connection with 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 generalized convex polytopes with positive singular curvature. This Hessian is shown to be equal to the Hessian of the volume of the dual generalized polyhedron. We prove the non-degeneracy by applying the technique used in the proof of Alexandrov-Fenchel inequality. Our construction of a convex polytope from a given metric is implemented in a computer program.

Nous présentons, dans cet article, une démonstration constructive du théorème d’Alexandrov sur l’existence d’un polytope convexe ayant une métrique donnée sur son bord. Le polytope est obtenu en déformant des polytopes convexes généralisés dont le bord est donné. Nous étudions l’espace des polytopes convexes généralisés et mettons en évidence une relation avec les triangulations de Delaunay pondérées des surfaces polyédrales. L’existence de la déformation est une conséquence de la non-dégénérescence du hessien de la courbure scalaire totale des polytopes convexes généralisés ayant leurs courbures singulières positives. Ce hessien se révèle être égal au hessien du volume du polyèdre généralisé dual. Nous démontrons la non-dégénérescence en appliquant la technique utilisée dans la preuve de l’inégalité d’Alexandrov-Fenchel. Notre construction d’un polytope convexe à partir d’une métrique donnée est mise en œuvre dans un programme informatique.

DOI : https://doi.org/10.5802/aif.2358
Classification:  52B10,  53C45,  52A39,  52C25
Keywords: Singular Euclidean metric, convex polytope, total scalar curvature
@article{AIF_2008__58_2_447_0,
     author = {Bobenko, Alexander I. and Izmestiev, Ivan},
     title = {Alexandrov's theorem, weighted Delaunay triangulations, and mixed volumes},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {58},
     number = {2},
     year = {2008},
     pages = {447-505},
     doi = {10.5802/aif.2358},
     mrnumber = {2410380},
     zbl = {1154.52005},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2008__58_2_447_0}
}
Bobenko, Alexander I.; Izmestiev, Ivan. Alexandrov’s theorem, weighted Delaunay triangulations, and mixed volumes. Annales de l'Institut Fourier, Volume 58 (2008) no. 2, pp. 447-505. doi : 10.5802/aif.2358. http://www.numdam.org/item/AIF_2008__58_2_447_0/

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