文摘
Let X be a real linear space, V be a nonempty subset of X and δ be a nonnegative real number. A function \({f : V \to \mathbb{R}}\) is said to be conditionally δ-midconvex provided \({f(\frac{x+y}{2}) \leq \frac{f(x) + f(y)}{2} + \delta}\) for every \({x, y \in V}\) such that \({\frac{x + y}{2} \in V}\) . We show that if V satisfies some reasonable assumptions, then for every bounded from above conditionally δ-midconvex function \({f : V \to \mathbb{R}}\) the following estimation holds: \({\sup f(V) \leq \sup f(ext \, V) + k (V)\delta}\) , where ext ? V denotes the set of all extremal points of V and k(V) is a respective constant depending on V.