Some Analytical Properties of γ-Convex Functions in Normed Linear Spaces

Abstract

For a fixed positive number γ, a real-valued function f defined on a convex subset D of a normed space X is said to be γ-convex if it satisfies the inequality

$$f(x^{\prime}_{0})+f(x^{\prime}_{1}) \leq f(x_0)+f(x_1), \quad \hbox{for } x^{\prime}_{i} \in \left[x_0,x_1\right], {\Vert {x^{\prime}_{i}} - {x^{}_{i}} \Vert} = \gamma,\quad i=0,1,$$

whenever x₀, x₁ ∈D and ${\Vert {x_{0}} - {x_{1}} \Vert} \geq \gamma$. This paper presents some results on the boundedness and continuity of γ-convex functions. For instance, (a) if there is some x_*∈D such that f is bounded below on D∩b̄(x_*,γ), then so it is on each bounded subset of D; (b) if f is bounded on some closed ball b̄(x_*,γ/2)⊂ 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 γ, then f is either locally bounded or nowhere locally bounded in the interior of D; (d) if D contains some open ball B(x_*,γ/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.