# 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.

## References

- 1.Hai, N.N., Phu, H.X.: Symmetrically
*γ*-convex functions. Optimization**46**, 1–23 (1999)CrossRefzbMATHMathSciNetGoogle Scholar - 2.Hartwig, H.: Local boundedness and continuity of generalized convex functions. Optimization
**26**, 1–13 (1992)CrossRefzbMATHMathSciNetGoogle Scholar - 3.Hu, T.C., Klee, V., Larman, D.: Optimization of globally convex functions. SIAM J. Control Optim.
**27**, 1026–1047 (1989)CrossRefzbMATHMathSciNetGoogle Scholar - 4.Phu, H.X.:
*γ*-subdifferential and*γ*-convexity of functions on the real line. Appl. Math. Optim.**27**, 145–160 (1993)CrossRefzbMATHMathSciNetGoogle Scholar - 5.Phu, H.X.:
*γ*-subdifferential and*γ*-convex functions on a normed space. J. Optim. Theory Appl.**85**, 649–676 (1995)CrossRefzbMATHMathSciNetGoogle Scholar - 6.Phu, H.X.: Six kinds of roughly convex functions. J. Optim. Theory Appl.
**92**, 357–375 (1997)CrossRefzbMATHMathSciNetGoogle Scholar - 7.Phu, H.X.: Outer Γ-convexity in vector spaces. Numer. Funct. Anal. Optim.
**25**, 835–854 (2008)CrossRefMathSciNetGoogle Scholar - 8.Phu, H.X.: Representation of bounded convex sets by rational convex hull of its
*γ*–extreme points. Numer. Funct. Anal. Optim.**15**, 915–920 (1994)CrossRefzbMATHMathSciNetGoogle Scholar - 9.Phu, H.X.: Supremizers of inner
*γ*-convex functions. Math. Methods Oper. Res.**67**, 207–222 (2008)CrossRefzbMATHMathSciNetGoogle Scholar - 10.Phu, H.X.: Outer
*γ*-convexity and inner*γ*-convexity of disturbed functions. Vietnam J. Math.**35**, 107–119 (2007)zbMATHMathSciNetGoogle Scholar - 11.Phu, H.X., An, P.T.: Outer
*γ*-convexity in normed linear spaces. Vietnam J. Math.**27**, 323–334 (1999)zbMATHMathSciNetGoogle Scholar - 12.Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)zbMATHGoogle Scholar
- 13.Söllner, B.: Eigenschaften
*γ*-grobkonvexer Mengen und Funktionen. Diplomarbeit. Universität Leipzig, Leipzig (1991)Google Scholar

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