New algorithms for convex interpolation

In various settings, from computer graphics to financial mathematics, it is necessary
to smoothly interpolate a convex curve from a set of data points. Standard interpolation
schemes do not respect convexity, and existing special purpose methods require arbitrary
choices and/or give interpolants that are very flat between data points. We consider a
broad set of spline-type schemes and show that convexity preservation requires the basic
spline to be infinitely differentiable but nonanalytic at its endpoints. Using such a
scheme - which essentially corresponds to building-in the possibility of "very flatness"
ab initio, rather than, say, enforcing it through extreme parameter choices  - gives far
more satisfactory numerical results.
Joint work with Eli Passov