## Abstract

A real-valued function *f* defined on a convex subset *D* of some normed linear space is said to be inner *γ*-convex w.r.t. some fixed roughness degree γ > 0 if there is a \(\nu \in]0, 1]\) such that \({\rm sup}_{\lambda\in[2,1+1/\nu]} \left(f((1 - \lambda)x_0 + \lambda x_1) - (1 - \lambda) f (x_0)-\right. \left.\lambda f(x_1)\right) \geq 0\) holds for all \(x_0, x_1 \in D\) satisfying ||*x*
_{0} − *x*
_{1}|| = ν*γ* and \(-(1/\nu)x_0+(1+1/\nu)x_1\in D\) . This kind of roughly generalized convex functions is introduced in order to get some properties similar to those of convex functions relative to their supremum. In this paper, numerous properties of their supremizers are given, i.e., of such \(x^* \in D\) satisfying lim \({\rm sup}_{x \to x^*}f(x) = {\rm sup}_{x \in D} f(x)\) . For instance, if an upper bounded and inner *γ*-convex function, which is defined on a convex and bounded subset *D* of some inner product space, has supremizers, then there exists a supremizer lying on the boundary of *D* relative to aff *D* or at a *γ*-extreme point of *D*, and if *D* is open relative to aff *D* or if dim *D* ≤ 2 then there is certainly a supremizer at a *γ*-extreme point of *D*. Another example is: if *D* is an affine set and \(f : D \to {\mathbb{R}}\) is inner *γ*-convex and bounded above, then \({\rm sup}_{x'\in \bar B(x,\gamma/2)\cap D}f(x')= \sup_{x'\in D}f(x')\) for all \(x \in D\) , and if 2 ≤ dim *D* < ∞ then each \(x \in D\) is a supremizer of *f*.

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## References

Hartwig H (1992). Local boundedness and continuity of generalized convex functions.

*Optimization*26: 1–13Hu TC, Klee V and Larman D (1989). Optimization of globally convex functions.

*SIAM J Control Optim*27: 1026–1047Phu HX (1993).

*γ*-Subdifferential and*γ*-convexity of functions on the real line.*Appl Math Optim*27: 145–160Phu HX (1994). Representation of bounded convex sets by rational convex hull of its

*γ*-extreme points.*Numer Funct Anal Optim*15: 915–920Phu HX (1995).

*γ*-Subdifferential and*γ*-convex functions on a normed space.*J Optim Theory Appl*85: 649–676Phu HX (1997). Six kinds of roughly convex functions.

*J Optim Theory Appl*92: 357–375Phu HX (2007a) Inner

*γ*-convex functions in normed spaces, E-Preprint 2007/01/01, Hanoi Institute of MathematicsPhu HX (2007b). Outer

*γ*-convexity and inner*γ*-convexity of disturbed functions.*Vietnam J Math*35: 107–119Rockafellar RT (1970). Convex analysis. Princeton University Press, Princeton

Söllner B (1991) Eigenschaften

*γ*-grobkonvexer Mengen und Funktionen. Diplomarbeit, Universität Leipzig

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Phu, H.X. Supremizers of inner *γ*-convex functions.
*Math Meth Oper Res* **67**, 207–222 (2008). https://doi.org/10.1007/s00186-007-0187-4

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DOI: https://doi.org/10.1007/s00186-007-0187-4