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.
@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},
volume = {58},
year = {2008},
pages = {447-505},
doi = {10.5802/aif.2358},
zbl = {1154.52005},
mrnumber = {2410380},
language = {en},
url = {http://dml.mathdoc.fr/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, Tome 58 (2008) pp. 447-505. doi : 10.5802/aif.2358. http://gdmltest.u-ga.fr/item/AIF_2008__58_2_447_0/
[1] On the theory of mixed volumes II, Mat. Sbornik, Tome 44 (1937), pp. 1205-1238
[2] Existence of a convex polyhedron and of a convex surface with a given metric, Mat. Sbornik, N. Ser., Tome 11(53) (1942), pp. 15-65 ((Russian. English summary)) | MR 13540 | Zbl 0061.37603
[3] Convex polyhedra, Springer-Verlag, Berlin, Springer Monographs in Mathematics (2005) (Translated from the 1950 Russian edition) | MR 2127379 | Zbl 1067.52011
[4] Voronoi diagrams, Handbook of computational geometry, North-Holland, Amsterdam (2000), pp. 201-290 | MR 1746678 | Zbl 0995.65024
[5] Ein Beweis für die Unverbiegbarkeit geschlossener konvexer Flächen., Nachr. Ges. Wiss. Göttingen (1912), pp. 607-610
[6] Über die Verwirklichung einer geschlossenen Fläche mit vorgeschriebenem Bogenelement im Euklidischen Raum, Sitzungsber. Bayer. Akad. Wiss., Math.-Naturwiss. Abt., Tome No.2 (1937), p. 229-230 | Zbl 0018.23501
[7] A discrete Laplace-Beltrami operator for simplicial surfaces (arXiv:math.DG/0503219, to appear in Discr. Comp. Geom.)
[8] Singular Euclidean structures on surfaces, J. London Math. Soc. (2), Tome 44 (1991) no. 3, pp. 553-565 | Article | MR 1149015 | Zbl 0748.57003
[9] Über die Starrheit konvexer Polyeder., Math. Ann., Tome 77 (1916), pp. 466-473 | Article | MR 1511873
[10] Geometry and topology for mesh generation, Cambridge University Press, Cambridge, Cambridge Monographs on Applied and Computational Mathematics, Tome 7 (2001) | MR 1833977 | Zbl 0981.65028
[11] Rigidity and polynomial invariants of convex polytopes, Duke Math. J., Tome 129 (2005) no. 2, pp. 371-404 | Article | MR 2165546 | Zbl 1081.52012
[12] Polyhedral realization of hyperbolic metrics with conical singularities on compact surfaces, Ann. Inst. Fourier (Grenoble), Tome 57 (2007) no. 1, pp. 163-195 | Article | Numdam | MR 2313089
[13] Rigidity and the Alexandrov-Fenchel inequality., Monatsh. Math., Tome 113 (1992) no. 1, pp. 1-22 | Article | MR 1149057 | Zbl 0765.52017
[14] Voronoi diagrams and Delaunay triangulations, Handbook of discrete and computational geometry, CRC, Boca Raton, FL (CRC Press Ser. Discrete Math. Appl.) (1997), pp. 377-388 | MR 1730176 | Zbl 0907.68190
[15] Discriminants, resultants, and multidimensional determinants, Birkhäuser Boston Inc., Boston, MA, Mathematics: Theory & Applications (1994) | MR 1264417 | Zbl 0827.14036
[16] Geometric triangulations and discrete Laplacians on manifolds, arXiv:math.MG/0508188
[17] Voronoi diagrams on piecewise flat surfaces and an application to biological growth, Theoret. Comput. Sci., Tome 263 (2001) no. 1-2, pp. 263-274 (Combinatorics and computer science (Palaiseau, 1997)) | Article | MR 1846934 | Zbl 0974.68222
[18] Beweis zweier Sätze über die Bestimmung von Ovaloiden durch das Krümmungsmass oder die mittlere Krümmung für jede Normalenrichtung., Nachr. Ges. Wiss. Göttingen (1899), pp. 134-142
[19] The Schläfli differential equality, Collected papers, Publish or Perish Inc., Houston, TX, Tome 1 (1994), pp. x+295
[20] Allgemeine Lehrsätze über die konvexen Polyeder, Nachr. Ges. Wiss. Göttingen (1897), pp. 198-219
[21] The Weyl and Minkowski problems in differential geometry in the large, Comm. Pure Appl. Math., Tome 6 (1953), pp. 337-394 | Article | MR 58265 | Zbl 0051.12402
[22] Rigidity and polynomial invariants of convex polytopes, Sib. Math. J., Tome 47 (2006) no. 5, pp. 859-864 | MR 2265287 | Zbl 1150.52311
[23] General relativity without coordinates, Nuovo Cimento, Tome 19 (1961), pp. 558-571 | Article | MR 127372
[24] Euclidean structures on simplicial surfaces and hyperbolic volume, Ann. of Math. (2), Tome 139 (1994) no. 3, pp. 553-580 | Article | MR 1283870 | Zbl 0823.52009
[25] Circle patterns on singular surfaces, arXiv:math.DG/0601531
[26] Small deformations of polygons and polyhedra, Trans. Amer. Math. Soc., Tome 359 (2007) no. 5, p. 2155-2189 (electronic) | Article | MR 2276616 | Zbl 05120631
[27] Convex bodies: the Brunn-Minkowski theory, Cambridge University Press, Cambridge, Encyclopedia of Mathematics and Its Applications, Tome 44 (1993) | MR 1216521 | Zbl 0798.52001
[28] A variational principle for weighted Delaunay triangulations and hyperideal polyhedra, arXiv:math/0603097 (to appear in J. Diff. Geom.)
[29] An estimate for the deformation of a convex surface in dependence on the variation of its intrinsic metric, Ukrain. Geometr. Sb., Tome 5–6 (1968), pp. 44-69 | MR 283734 | Zbl 0207.20801
[30] Existence of a convex polyhedron with prescribed development, Taškent. Gos. Ped. Inst. Učen. Zap., Tome 85 (1971), p. 3-54, 83 ((Russian)) | MR 328776
[31] Über die Bestimmung einer geschlossenen konvexen Fläche durch ihr Linienelement, Zürich. Naturf. Ges., Tome 61 (1916), pp. 40-72
[32] Lectures on polytopes, Springer-Verlag, Berlin, Graduate Texts in Mathematics, Tome 152 (1995) | MR 1311028 | Zbl 0823.52002