Local boundedness and continuity of generalized convex functions

This article deals with generalizations of the usual convexity of real-valued functions in such a manner that “convex” is extended to “-convex” and -convexity is required only on straight lines with directions from a given cone K. Under certain assumptions on the generating family and on K, for functions of such kind (called -convex on K-lines) local boundedness and continuity properties are obtained. The main results are applied to a number of examples. In particular, Morrey’s rank 1 convexity and a special type of “rough convexity” are considered