WEAK CONVERGENCE OF INTEGRAL CURVATURES OF CONVEX SURFACES
Ключевые слова:
convex surface, convex surfaces in Euclidean space, Monge-Ampère equation, the cone of convex surfaces in the space of continuous functions, conditional curvature, integral curvature, restoration of surface.Аннотация
The article contains a concentrated analysis of the existing information on the main problems of the
theory of convex surfaces and differential geometry “in general” and is devoted to the problems of the reconstructing
convex surfaces from the information about their curvature studied by the topological methods of the functional
analysis.
The class of smooth surfaces in a bounded convex domain GE2 is considered. The concept of the R-area of a
normal image is set forth. In the class K+(G), the Monge–Ampere equation is considered.
The paper considers the integrals of transverse Minkowski measures associated with the parallel surfaces. If the
surface Φ is given by the explicit equation z = f (x, y), then for the integral curvatures of this surface transferred to
the plane E2, the inequalities. Of these inequalities, inequalities follow
which are used in reasoning. We prove the weak convergence of the integral curvatures of the convex surfaces. The
result obtained in the form of a theorem plays an important role in the proofs of the theorems on the existence of a
convex hypersurface with a given combination of the integral conditional curvatures. For the first time, the
conditional curvatures are taken in the most general form, as a given function of the integral conditional curvatures
of the various orders. The integrand functions are the product of the continuous functions and the integral curvatures
of the various orders.
