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

@misc{citeulike:848575, 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.}, archivePrefix = {arXiv}, author = {Bobenko, Alexander I. and Izmestiev, Ivan}, citeulike-article-id = {848575}, citeulike-linkout-0 = {http://arxiv.org/abs/math.DG/0609447}, citeulike-linkout-1 = {http://arxiv.org/pdf/math.DG/0609447}, day = {15}, eprint = {math.DG/0609447}, keywords = {aleksandrov, algorithm, assignment, convex, convex-polyhedron}, month = {Sep}, posted-at = {2006-09-18 12:15:34}, priority = {2}, title = {Alexandrov's theorem, weighted Delaunay triangulations, and mixed volumes}, url = {http://arxiv.org/abs/math.DG/0609447}, year = {2006} }

TY - ELEC ID - citeulike:848575 L3 - citeulike-article-id:848575 N2 - 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. TI - Alexandrov's theorem, weighted Delaunay triangulations, and mixed volumes KW - aleksandrov KW - algorithm KW - assignment KW - convex KW - convex-polyhedron AU - Bobenko, Alexander AU - Izmestiev, Ivan PY - 2006/Sep/15/ UR - http://arxiv.org/abs/math.DG/0609447 ER -