In this paper, we introduce the concept of convex 2-metric spaces and present a generalization of Banach contraction principle in this newly defined space. Our result is a generalization of some well known results of 2-metric spaces. In 1970 Takahashi [4] introduced the notion of convex metric spaces and studied some fixed point theorems for non-expansive mappings in this space. Gahler [1] introduced the concept of 2-metric space alike to metric space. The purpose of this paper is to introduce the concept of convex 2-metric space analogue to convex metric space and obtain a generalization of Banach contraction principle in this space.