### Journals Information

**
Mathematics and Statistics Vol. 11(3), pp. 509 - 515 DOI: 10.13189/ms.2023.110306 Reprint (PDF) (241Kb) **

## Existence and Uniqueness of Polyhedra with Given Values of the Conditional Curvature at the Vertices

**Anvarjon Sharipov ^{*}, Mukhamedali Keunimjaev **

Department of Geometry and Topology, Faculty of Mathematics, National University of Uzbekistan, Uzbekistan

**ABSTRACT**

The theory of polyhedra and the geometric methods associated with it are not only interesting in their own right but also have a wide outlet in the general theory of surfaces. Certainly, it is only sometimes possible to obtain the corresponding theorem on surfaces from the theorem on polyhedra by passing to the limit. Still, the theorems on polyhedra give directions for searching for the related theorems on surfaces. In the case of polyhedra, the elementary-geometric basis of more general results is revealed. In the present paper, we study polyhedra of a particular class, i.e., without edges and reference planes perpendicular to a given direction. This work is a logical continuation of the author's work, in which an invariant of convex polyhedra isometric on sections was found. The concept of isometry of surfaces and the concept of isometry on sections of surfaces differ from each other, examples of isometric surfaces that are not isometric on sections and examples of non-isometric surfaces that are isometric on sections. However, they have non-empty intersections, i.e., some surfaces are both isometric and non-isometric on sections. In this paper, we prove the positive definiteness of the found invariant. Further, conditional external curvature is introduced for "basic" sets, open faces, edges, and vertices. It is proved that the conditional curvature of the polyhedral angle considered is monotonicity and positive definiteness. At the end of the article, the problem of the existence and uniqueness of convex polyhedra with given values of conditional curvatures at the vertices is solved.

**KEYWORDS**

Isometry on Section, The Conditional Total Angel, Invariant, Conditional Curvature

**Cite This Paper in IEEE or APA Citation Styles**

(a). IEEE Format:

[1] Anvarjon Sharipov , Mukhamedali Keunimjaev , "Existence and Uniqueness of Polyhedra with Given Values of the Conditional Curvature at the Vertices," Mathematics and Statistics, Vol. 11, No. 3, pp. 509 - 515, 2023. DOI: 10.13189/ms.2023.110306.

(b). APA Format:

Anvarjon Sharipov , Mukhamedali Keunimjaev (2023). Existence and Uniqueness of Polyhedra with Given Values of the Conditional Curvature at the Vertices. Mathematics and Statistics, 11(3), 509 - 515. DOI: 10.13189/ms.2023.110306.