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 <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$\nu \in]0, 1]$$</EquationSource> </InlineEquation> such that <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$${\rm sup}_{\lambda\in[2,1+1/\nu]} \left(f((1 - \lambda)x_0 + \lambda x_1) - (1 - \lambda) f (x_0)-\right. \left.\lambda f(x_1)\right) \geq 0$$</EquationSource> </InlineEquation> holds for all <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$x_0, x_1 \in D$$</EquationSource> </InlineEquation> satisfying ||x <Subscript>0</Subscript> − x <Subscript>1</Subscript>|| = νγ and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$-(1/\nu)x_0+(1+1/\nu)x_1\in D$$</EquationSource> </InlineEquation> . 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 <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$x^* \in D$$</EquationSource> </InlineEquation> satisfying lim <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$${\rm sup}_{x \to x^*}f(x)={\rm sup}_{x \in D} f(x)$$</EquationSource> </InlineEquation> . 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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$f : D \to {\mathbb{R}}$$</EquationSource> </InlineEquation> is inner γ-convex and bounded above, then <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$${\rm sup}_{x'\in \bar B(x,\gamma/2)\cap D}f(x')= \sup_{x'\in D}f(x')$$</EquationSource> </InlineEquation> for all <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$$x \in D$$</EquationSource> </InlineEquation> , and if 2 ≤ dim D > ∞ then each <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$$x \in D$$</EquationSource> </InlineEquation> is a supremizer of f. Copyright Springer-Verlag 2008
