arXiv subjects

Ivan Izmestiev

Publications and source records attributed to Ivan Izmestiev.

39 records · Page 3Linked to original sources

arXiv2006

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

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.

math.DG↗

arXiv2004

On the hull numbers of torus links

The n-th hull of a union of curves in R^3 is the set of points with the property: Any plane passing through the point intersects the curves at least 2n times. The hull number u(L) of a link L is defined as the minimum number of non-empty hulls a representative of L can have. We show that the hull numbers of torus links are smaller than expected, but still large. In particular, for a link of type (p,p) hull number equals 3p/5, and in general for the type (p,q) with q>p it is greater than or equal to p/2.

math.GT↗

arXiv2002

Branched Coverings, Triangulations, and 3-Manifolds

A canonical branched covering over each sufficiently good simplicial complex is constructed. Its structure depends on the combinatorial type of the complex. In this way, each closed orientable 3-manifold arises as a branched covering over the 3-sphere from some triangulation of S^3. This result is related to a theorem of Hilden and Montesinos. The branched coverings introduced admit a rich theory in which the group of projectivities plays a central role.

math.GT↗