Conditionally approximately convex functions

Abstract

Let X be a real normed space, V be a subset of X and α: [0, ∞) → [0, ∞] be a nondecreasing function. We say that a function f : V → [−∞, ∞] is conditionally α-convex if for each convex combination ∑i=0ntivi$\sum\nolimits_{i = 0}^n {t_i v_i }$ of elements from V such that ∑i=0ntivi∈V$\sum\nolimits_{i = 0}^n {t_i v_i \in V}$ , the following inequality holds true f(∑i=0ntivi)≤∑i=0ntif(vi)+α(max⁡i∈{0,…,n}ti‖vi−∑i=0ntivi‖).$$f\left( {\sum\limits_{i = 0}^n {t_i v_i } } \right) \le \sum\limits_{i = 0}^n {t_i f(v_i )} + \alpha (\mathop {\max }\limits_{i \in \{ 0, \ldots ,n\} } \left. {t_i } \right\|v_i - \sum\limits_{i = 0}^n {t_i v_i } \left\| ) \right..$$

Keywords

• convex function
• approximately convex function
• conditionally convex function
• 26A51
• 39B62