m-convex (m-cv) functions
Abstract
The theory of m-convex (m -cv) functions is a new direction in the theory of real geometry.However, for m = 1 this class coincides with the class of convex functions, and for m = n it coincides with the class of subharmonic functions, which, as is known, have been well studied (A.Aleksandrov, I.Bakelman, A.Pogorelov, N.Ivochkina, I.Privalov, etc.) The definition of m -cv functions for 1 < m < n has a very different nature, which uses high-order Hessians.Functions for such m have been considered in a series of works by N.Trudinger, X.Wang and others.In this article, we establish a connection between m-convex functions and strongly m-subharmonic (sh m ) functions and, using the well-known properties of sh m functions, we prove a number of important properties of the class of m -cv functions.