[Théorème d’Alexandrov, triangulations de Delaunay pondérées et volumes mixtes]
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.
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.
Keywords: Singular Euclidean metric, convex polytope, total scalar curvature
Mot clés : métrique singulière euclidienne, polytope convexe, courbure scalaire totale
@article{AIF_2008__58_2_447_0,
author = {Bobenko, Alexander I. and Izmestiev, Ivan},
title = {Alexandrov{\textquoteright}s theorem, weighted {Delaunay} triangulations, and mixed volumes},
journal = {Annales de l'Institut Fourier},
pages = {447--505},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {58},
number = {2},
year = {2008},
doi = {10.5802/aif.2358},
zbl = {1154.52005},
mrnumber = {2410380},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/aif.2358/}
}
TY - JOUR AU - Bobenko, Alexander I. AU - Izmestiev, Ivan TI - Alexandrov’s theorem, weighted Delaunay triangulations, and mixed volumes JO - Annales de l'Institut Fourier PY - 2008 SP - 447 EP - 505 VL - 58 IS - 2 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.2358/ DO - 10.5802/aif.2358 LA - en ID - AIF_2008__58_2_447_0 ER -
%0 Journal Article %A Bobenko, Alexander I. %A Izmestiev, Ivan %T Alexandrov’s theorem, weighted Delaunay triangulations, and mixed volumes %J Annales de l'Institut Fourier %D 2008 %P 447-505 %V 58 %N 2 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.2358/ %R 10.5802/aif.2358 %G en %F 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, Tome 58 (2008) no. 2, pp. 447-505. doi : 10.5802/aif.2358. http://archive.numdam.org/articles/10.5802/aif.2358/
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