Supremizers of inner γ-convex functions

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||  =  νγ 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.