Interpolation of Convex Scattered Data in R3 Using Edge Convex Minimum Lp-Norm Networks, 1 < p < ∞

We consider the extremal problem of interpolation of scattered data in R-3 by smooth curve networks with minimal L-P-norm of the second derivative for 1 < p < infinity. The problem for p = 2 was set and solved by Nielson [7]. Andersson et al. [1] gave a new proof of Nielson's result by using a different approach. It allowed them to set and solve the constrained extremal problem of interpolation of convex scattered data in R-3 by minimum L-2-norm networks that are convex along the edges of an associated triangulation. Partial results for the unconstrained and the constrained problems were announced without proof in [8]. The unconstrained problem for 1 < p < infinity was fully solved in [10]. Here we present complete characterization of the solution to the constrained problem for 1 < p < infinity.