Based on the analysis of the classifical 4 point linear interpolatory subdivision scheme introduced by Dyn, a functional nonlinear discrete subdivision scheme is presented. This scheme has the preserving convexity property, i.e., for any given convex discrete data, when some conditions are satisfied, the subdivision polygon curve produced in any step by this scheme is convex, so the limit curve is also convex. Some numerical examples show that the limit curves are fractal like when the smooth condition is not satisfied.