Some analytical properties of γ-convex functions in normed linear spaces

For a fixed positive number gamma, a real-valued function f defined on a convex subset D of a normed space X is said to be gamma-convex if it satisfies the inequality f (x'(0)) + f (x'(1)) <= f (x(0)) + f (x(1)), for x'(i) is an element of[x(0), x(1)], parallel to x'(i) - x(i)parallel to = gamma, i = 0, 1, whenever x(0), x(1) is an element of D and parallel to x(0) - x(1)parallel to >= gamma. This paper presents some results on the boundedness and continuity of gamma-convex functions. For instance, (a) if there is some x(*) is an element of D such that f is bounded below on D boolean AND(B) over bar (x(*), gamma), then so it is on each bounded subset of D; (b) if f is bounded on some closed ball (B) over bar( x*, gamma/2) subset of D and D' is a closed bounded subset of D, then f is bounded on D' iff it is bounded above on the boundary of D'; ( c) if dim X> 1 and the interior of D contains a closed ball of radius., then f is either locally bounded or nowhere locally bounded in the interior of D; (d) if D contains some open ball B(x(*), gamma/2) in which f has at most countably many discontinuities, then the set of all points at which f is continuous is dense in D.