arXiv subjects

Victor Alexandrov

Publications and source records attributed to Victor Alexandrov.

41 records · Page 3Linked to original sources

arXiv2005

Blaschke addition and convex polyhedra

This is an extended version of a talk on October 4, 2004 at the research seminar ``Differential geometry and applications'' (headed by Academician A. T. Fomenko) at Moscow State University. The paper contains an overview of available (but far from well-known) results about the Blaschke addition of convex bodies, some new theorems on the monotonicity of the volume of convex bodies (in particular, convex polyhedra with parallel faces) as well as description of a software for visualization of polyhedra with prescribed outward normals and face areas.

math.MG↗

arXiv2002

Minkowski-type and Alexandrov-type theorems for polyhedral herissons

Classical H.Minkowski theorems on existence and uniqueness of convex polyhedra with prescribed directions and areas of faces as well as the well-known generalization of H.Minkowski uniqueness theorem due to A.D.Alexandrov are extended to a class of nonconvex polyhedra which are called polyhedral herissons and may be described as polyhedra with injective spherical image.

math.MG↗

arXiv2001

Flexible polyhedra in the Minkowski 3-space

Given a polyhedral surface, assume that it is prohibited to change the shape and size of any face but it is permissible to change the dihedral angles between the faces. A polyhedral surface is said to be flexible if it is possible to change its shape under the above restrictions. We prove that flexible polyhedral surfaces without boundary do exist in the Minkowski 3-space and each of them preserves the (inclosed) volume and the (total) mean curvature during a flex. To prove the latter result, we introduce the notion of the angle between two arbitrary non-null nonzero vectors in the Minkowski plane. The latter notion may be of some independent interest.

math.MG↗

arXiv2000

Implicit Function Theorem for systems of polynomial equations with vanishing Jacobian and its application to flexible polyhedra and frameworks

We study the existence problem for a local implicit function determined by a system of nonlinear algebraic equations in the particular case when the determinant of its Jacobian matrix vanishes at the point under consideration. We present a system of sufficient conditions that implies existence of a local implicit function as well as another system of sufficient conditions that guarantees absence of a local implicit function. The results obtained are applied to proving new and classical results on flexibility and rigidity of polyhedra and frameworks.

math.MG↗

arXiv1999

Higher dimensional flexible polyhedra

In the previous version of the paper it was announced that ``sphere homeomorphic flexible polyhedra (with self intersections) do really exist in n-dimensional Euclidean, Lobachevskij and spherical spaces for each $n\geq 3$.'' Now the paper has been withdrawn by the author, due a crucial error in the proof of the main theorem.

math.MG↗