ALEXANDROV'S THEOREM, WEIGHTED DELAUNAY TRIANGULATIONS, AND MIXED VOLUMES

By Alexander I. Bobenko, Ivan Izmestiev. 2006

101 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Computer Graphics and Computer-Aided Design, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/54f4accf9ec6543d78c76cb9bb62c031a63d1027

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

1. David Glickenstein
2. Boris A. Springborn
3. Alexander I. Bobenko
4. Ulrich Pinkall
5. Huabin Ge
6. Feng Luo
7. Xu Xu

In Computer Graphics and Computer-Aided Design

1. David Glickenstein
2. Alexander I. Bobenko
3. Boris A. Springborn
4. Ulrich Pinkall
5. Xu Xu
6. Feng Luo
7. Ren Guo
8. Ivan Izmestiev

In Geometry and Topology

1. David Glickenstein
2. Boris A. Springborn
3. Alexander I. Bobenko
4. Ulrich Pinkall
5. Feng Luo
6. Xu Xu
7. Huabin Ge
8. Wenshuai Jiang

Citations in the corpus

• A variational principle for weighted Delaunay triangulations and hyperideal polyhedra (2006, background): “Another interesting question is this: Can one formulate a dual variational principle for weighted Delaunay triangulations, with an explicit formula for the functional, where the variables are circle radii and inversive distances?”
• Discrete conformal variations and scalar curvature on piecewise flat two and three dimensional manifolds (2009, methodology): “Recently, the functional was used to provide a constructive proof of Alexandrov’s theorem that a surface with positive curvature is the boundary of a polytope [4].”
• Discrete conformal maps and ideal hyperbolic polyhedra (2010, background): “tive proof of Alexandrov’s classical polyhedral realization theorem [3].”