# Inner *γ*-Convex Functions in Normed Linear Spaces

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## Abstract

A real-valued function holds for all

*f*defined on a convex subset*D*of some normed linear space*X*is said to be inner*γ*-convex w.r.t. some fixed roughness degree*γ*> 0 if there is a*ν*∈ [0, 1] such that$$\sup \limits _{\lambda \in [2, 1+1/\nu ]} \left (f((1-\lambda )x_{0}+\lambda x_{1})- (1-\lambda )f(x_{0})- \lambda f(x_{1})\right )\geq 0 $$

*x*_{0},*x*_{1}∈*D*satisfying ∥*x*_{0}−*x*_{1}∥ =*ν**γ*and −(1/*ν*)*x*_{0}+ (1 + 1/*ν*)*x*_{1}∈*D*. The requirement of this kind of roughly generalized convex functions is very weak; nevertheless, they also possess properties similar to those of convex functions relative to their supremum. For instance, if an inner*γ*-convex function defined on some bounded convex subset*D*of an inner product space attains its maximum, then it has maximizers at some strictly*γ*-extreme points of*D*. In this paper, some sufficient conditions and examples for*γ*-convex functions and several properties relative to the location of their maximizers are given.## Keywords

Generalized convexity Rough convexity Inner*γ*-convex function

*γ*-Extreme point

## Mathematics Subject Classification (2010)

52A01 52A41## Notes

### Acknowledgments

The financial support offered by Vietnam’s National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2014.28 is acknowledged.

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© Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2015