  We consider spline estimates which preserve prescribed piecewise convex
properties of the unknown function. A robust version of the penalized
likelihood is given and shown to correspond to a variable halfwidth kernel
smoother where the halfwidth adaptively decreases in regions of rapid change of
the unknown function. When the convexity change points are prescribed, we
derive representation results and smoothness properties of the estimates. A
dual formulation is given which reduces the estimate is reduced to a finite
dimensional convex optimization in the dual space.
