The significance of convexity in many areas of geometry is well- known; especially the solutions of some important PDEs, such as Abreu's and Monge-Amper equations, should be convex. In this lecture, by introducing some of the proximal calculus tools, I will talk about separation theorem for convex subsets of Riemannian manifolds. Moreover, I will explain why convexity of distance function $d_S$ to a closed convex subset $S$ of the Riemannian manifold with positive sectional curvature fails in some cases and by imposing some restriction on $S$, independently of curvature of M, the convexity of $d_S$ will be obtained.