Outer Γ-convex Functions on a Normed Space

For some given positive γ, a function f is called outer γ-convex if it satisfies the Jensen inequality f(zi) ≤ (1 − λi)f(x0) + λif(x1) for some z0 : = x0, z1, ..., zk : = x1 ∈ [x0, x1] satisfying ‖zi − zi+1‖ ≤ γ, where λi : = ‖x0 − zi‖/‖x0 − x1‖, i = 1, 2, ..., k − 1. Though the Jensen inequality is only required to hold true at some points (although the location of these points is uncertain) on the segment [x0, x1], 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 γ-convex function attains −∞ at some point then this propagates to other points, and under some assumptions, a function is outer γ-convex iff its epigraph is an outer γ-convex set.