Geometric Convex Function Based on Geometric Convex Set

Abstract

Geometric convex function is a parallel concept with convex function and is given by a transformation or an inequality, which generalizes the theory and method of control inequality for convex function. With the concept of geometric convex set being put forward, the properties of geometric convex function based on geometric convex set have been studied. But the properties of geometric convex set and geometric convex function are not perfect. In this paper, we further discuss the properties of geometric convex set and geometric convex function by literature research, analysis, synthesis, induction and deduction. Firstly, based on the concept of geometric convex set, the geometric convex hull and geometric convex combination of a set are defined, and an equivalent condition for judging geometric convex set and an example of geometric convex combination are given. Then, the equivalent definition of geometric convex function is given by using the epigraph, and a judgment theorem of geometric convex function is obtained. In addition, the geometric convex hull of the function is defined, and another judgment theorem of geometric convex function is also obtained by using geometric convex hull of the function. Finally, the invariance of geometric convexity under several operations and the local and global properties of geometric convex functions are studied. The results of this paper will further enrich the theory of geometric convexity.