It is known that the points of a convex surface are divided into three classes: regular points, edge points, and vertices. There exists a class of surfaces that have a vertex at a single point, with all other points being regular. Such surfaces are called single-vertex surfaces, and the article provides some properties of the external curvature of such surfaces. Furthermore, the existence and uniqueness of a single-vertex surface have been proven, where the boundary consists of some closed spatial curve, and the external curvature is equal to a function defined on all Borel sets, given in advance. In this case, certain necessary conditions are imposed on the external curvature function.
Donyorbek Tillayev (Tue,) studied this question.