In complex geometry and analysis, the notion of convexity and its generalizations play an important role in understanding function behavior. In Mathematics, complex geometry is the studyof Complex manifolds and functions of many complex variables Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex In Mathematics, a real-valued function f defined on an interval (or on any Convex subset of some Vector space) is called convex The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function Examples of classes of functions with a rich structure are, in addition to the convex functions, the subharmonic functions and the plurisubharmonic functions. In Mathematics, subharmonic and superharmonic functions are important classes of functions used extensively in Partial differential equations In Mathematics, plurisubharmonic functions form an important class of functions used in Complex analysis. Geometrically, these classes of functions correspond to convex domains and pseudoconvex domains, but there are also other types of domains, for instance lineally convex domains which can be generalized using convex analysis. Convex analysis is the branch of Mathematics devoted to the study of properties of Convex functions and Convex sets often with applications in Convex A great deal is already known about these domains, but there remain some fascinating, unsolved problems. This theme is mainly theoretical, but there are computational aspects of the domains studied, and these computational aspects are certainly worthy of further study.