26A51, 26B25, 52A41.

For some given positive , a function is called outer -convex if it satisfies the Jensen inequality for some satisfying , where . Though the Jensen inequality is only required to hold true at some points (although the location of these points is uncertain) on the segment , such a function has many interesting properties similar to those of classical convex functions. Among others it is shown that, if the infimum limit of an outer -convex function attains at some point then this propagates to other points, and under some assumptions, a function is outer -convex iff its epigraph is an outer -convex set.