Abstract
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.
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Chudziak, J., Tabor, J. & Tabor, J. Conditionally δ-midconvex functions. Aequat. Math. 89, 981–990 (2015). https://doi.org/10.1007/s00010-014-0304-4
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DOI: https://doi.org/10.1007/s00010-014-0304-4