Vietnam Journal of Mathematics

, Volume 43, Issue 2, pp 487–500 | Cite as

Inner γ-Convex Functions in Normed Linear Spaces

  • Hoang Xuan PhuEmail author


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
$$\sup \limits _{\lambda \in [2, 1+1/\nu ]} \left (f((1-\lambda )x_{0}+\lambda x_{1})- (1-\lambda )f(x_{0})- \lambda f(x_{1})\right )\geq 0 $$
holds for all x 0, x 1D satisfying ∥x 0x 1∥ = ν γ and −(1/ν)x 0 + (1 + 1/ν)x 1D. 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.


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

Mathematics Subject Classification (2010)

52A01 52A41 



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


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

Copyright information

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

Authors and Affiliations

  1. 1.Institute of Mathematics, Vietnam Academy of Science and TechnologyCau Giay DistrictVietnam

Personalised recommendations