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Inner γ-Convex Functions in Normed Linear Spaces


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.

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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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Correspondence to Hoang Xuan Phu.

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

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Phu, H.X. Inner γ-Convex Functions in Normed Linear Spaces. Vietnam J. Math. 43, 487–500 (2015).

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

Mathematics Subject Classification (2010)

  • 52A01
  • 52A41