In the current paper we discussed some applications of fixed-point theorem especially convexity structure of fixed-point theorem along with pinot's formation there are some conditions on which the convexity structure was developed through the Kirks theorem which depends on metric space of fixed-point theorem and also showing some type of compactness and the characteristics of normal space we focused on some theorem in which we saw that how fixed point follows the property of convexity and how it became normal and compact studied the property of expansive mapping has fixed point Hence various steps in unmetrical analysis and its theory of approximations which is in successive form of fixed-point theorem and it mapping in this paper We give a proper framework to the concept of convex of metric space through the way of convexity structure by using this application we develop new dimension in metric space.