## Abstract

A real-valued function *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

holds for all *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.

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## 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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## Additional information

Dedicated to Professor Nguyen Khoa Son on the occasion of his 65th birthday.

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### Cite this article

Phu, H.X. Inner *γ*-Convex Functions in Normed Linear Spaces.
*Vietnam J. Math.* **43**, 487–500 (2015). https://doi.org/10.1007/s10013-015-0125-3

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DOI: https://doi.org/10.1007/s10013-015-0125-3

### Keywords

- Generalized convexity
- Rough convexity
- Inner
*γ*-convex function -
*γ*-Extreme point

### Mathematics Subject Classification (2010)

- 52A01
- 52A41