Around A.d. Alexandrov’s Uniqueness Theorem for 3d Polytopes
نویسندگان
چکیده
Two dependent examples are presented. 1. An example of two different convex 3D polytopes such that for each pair of their parallel facets, the facets are different, and there exists a unique translation putting one facet inside the other. 2. An example of a pointed tiling of S generated by a Laman-plus-one graph which can be regularly triangulated without adding new vertices. The paper explores the relationship between the theory of pseudo triangulations and the theory of hyperbolic virtual polytopes.
منابع مشابه
An Illustrated Theory of Hyperbolic Virtual Polytopes
The paper gives an illustrated introduction to the theory of hyperbolic virtual polytopes and related counterexamples to A.D. Alexandrov’s conjecture.
متن کاملPointed Spherical Tilings and Hyperbolic Virtual Polytopes
The paper presents an introduction to the theory of hyperbolic virtual polytopes from the viewpoint of combinatorial rigidity theory. Namely, we give a shortcut for a reader who is acquainted with the notions of Laman graphs, 3D liftings and pointed tilings. From this viewpoint, a hyperbolic virtual polytope is a stressed pointed graph embedded in the sphere S. The advantage of such a presentat...
متن کاملA. D. Alexandrov’s Uniqueness Theorem for Convex Polytopes and its Refinements
In 1937, A. D. Alexandrov proved that if no parallel faces of two 3-dimensional convex polytopes can be placed strictly one into another via a translation, then the polytopes are translates of one another. The theory of hyperbolic virtual polytopes elucidates this theorem and suggests natural ways of its refinement. Namely, we present an example of two different 3-dimensional polytopes such tha...
متن کاملAn Extension of Alexandrov’s Theorem on Second Derivatives of Convex Functions
If f is a function of n variables that is locally uniformly approximable by a sequence of smooth functions satisfying local L1 bounds on the determinants of the minors of the Hessian, then f admits a second order Taylor expansion almost everywhere. This extends a classical theorem of A.D. Alexandrov, covering the special case in which f is locally convex.
متن کاملPlanar Pseudo-triangulations, Spherical Pseudo-tilings and Hyperbolic Virtual Polytopes
We wish to draw attention to an interesting and promising interaction of two theories. On the one hand, it is the theory of pseudo-triangulations which was useful for implicit solution of the carpenter’s rule problem and proved later to give a nice tool for graph embeddings. On the other hand, it is the theory of hyperbolic virtual polytopes which arose from an old uniqueness conjecture for con...
متن کامل