The Unified Description of Abstract Convexity Structures

The convexity of space is essential in nonlinear analysis, variational inequalities and optimization theory because it guarantees the existence and uniqueness of solutions to a certain extent. Because of its wide variety of applications, mathematicians have extensively promoted and researched convexity. This paper reviews some representative convexity structures and discusses their relations from their definitions, unifying them in the abstract convex structure. Moreover, applications of main convexity structures including KKM theory and fixed point theory will be reviewed.