Convexity preserving interpolatory subdivision schemes

We construct local subdivision schemes that interpolate functional univariate data and that preserve convexity. The resulting limit function of these schemes is continuous and convex for arbitrary convex data. Moreover this class of schemes is restricted to a subdivision scheme that generates a limit function that is convex and continuously differentiable for strictly convex data. The approximation order of this scheme is four. Some generalizations, such as tension control and piecewise convexity preservation, are briefly discussed.