Approximately Midconvex Functions

Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 52)

Abstract

In the paper we propose very general definition of approximate midconvexity. Let α: [0, ) → be a given function. Let X be a normed space and V a convex subset of X. A function f: V will be called α(⋅) - midconvex if
$$f\left (\frac{x + y} {2} \right ) \leq \frac{1} {2}f(x) + \frac{1} {2}f(y) + \alpha (\|x - y\|)\mbox{ for }x,y \in V.$$
The above definition simultaneously generalizes approximate and uniform midconvexities. We present several results concerning this notion.

Keywords

Approximately convex function Approximately midconvex function Semiconvex function 

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Institute of Computer ScienceJagiellonian UniversityKrakówPoland
  2. 2.Institute of MathematicsUniversity of RzeszówRzeszówPoland

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