Strictly and Roughly Convexlike Functions

A function is said to be strictly and roughly convexlike with respect to the roughness degree r > 0 (for short, strictly r-convexlike) provided that, for all x₀, x₁ D satisfying ||x₀ – x₁|| > r, there exists a ]0, 1[ such that.The most important property of strictly r-convexlike functions is that the diameter of the set of global minimizers is not greater than r. This property is needed in another paper for obtaining the rough stability of optimal solutions to nonconvex parametric optimization problems. Moreover, if f is supposed to be lower semicontinuous, then each r-local minimizer x*, defined byis a global minimizer of f. In this paper, necessary and sufficient conditions for a function to be strictly r-convexlike are stated. In particular, the class of strictly -convex functions is considered.