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

Alexander I. Bobenko[1]; Ivan Izmestiev[1]

  • [1] Technische Universität Berlin Institut für Mathematik Str. des 17. Juni 136 10623 Berlin (Germany)

Annales de l’institut Fourier (2008)

  • Volume: 58, Issue: 2, page 447-505
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Bobenko, Alexander I., and Izmestiev, Ivan. "Alexandrov’s theorem, weighted Delaunay triangulations, and mixed volumes." Annales de l’institut Fourier 58.2 (2008): 447-505. <http://eudml.org/doc/10322>.

@article{Bobenko2008,
abstract = {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.},
affiliation = {Technische Universität Berlin Institut für Mathematik Str. des 17. Juni 136 10623 Berlin (Germany); Technische Universität Berlin Institut für Mathematik Str. des 17. Juni 136 10623 Berlin (Germany)},
author = {Bobenko, Alexander I., Izmestiev, Ivan},
journal = {Annales de l’institut Fourier},
keywords = {Singular Euclidean metric; convex polytope; total scalar curvature; singular euclidean metric; Delaunay triangulation; generalized convex polytope; generalized dual; mixed volumes.},
language = {eng},
number = {2},
pages = {447-505},
publisher = {Association des Annales de l’institut Fourier},
title = {Alexandrov’s theorem, weighted Delaunay triangulations, and mixed volumes},
url = {http://eudml.org/doc/10322},
volume = {58},
year = {2008},
}

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
PB - Association des Annales de l’institut Fourier
VL - 58
IS - 2
SP - 447
EP - 505
AB - 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.
LA - eng
KW - Singular Euclidean metric; convex polytope; total scalar curvature; singular euclidean metric; Delaunay triangulation; generalized convex polytope; generalized dual; mixed volumes.
UR - http://eudml.org/doc/10322
ER -

References

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