Symmetricallyγ-Convex Functions
N. N Hai, Hoàng Xuân Phú · Optimization · 1999
Given γ>0, a real-valued function f defined on a noempty convex set D of a normed space X is said to be symmetrically γ-convex iff for all x oand X 1∊D satisfying , the Jensen inequality is fulfilled for and for . Though the Jensen inequality is only required to hold true at two points and on the segment [x 0,x 1] satisfying , such a function has interesting properties similar to those of classical convex functions. For instance, a symmetrically γ-convex function from a finite-dimensional normed space X is locally Lipschitzian at each γ-interior point of D, i.e., at each point x∊D for which there is some ε>0 such that implies . This and other properties of this function class are established in our paper