Extremal interpolation of convex scattered data in R³ by smooth edge convex minimum L∞‐norm networks: Characterization and solution

We consider the extremal problem of interpolation of convex scattered data in by smooth edge convex curve networks with minimal ‐norm of the second derivative for . The problem for was set and solved by Andersson et al. (1995). Vlachkova (2019) extended the results of Andersson et al. (1995) and solved the problem for . The minimum edge convex ‐norm network for is obtained from the solution to a system of nonlinear equations with coefficients determined by the data. The solution in the case is unique for strictly convex data. The corresponding extremal problem for remained open. The case is of particular interest in the context of applications since it has a solution which is a smooth curve network consisting of quadratic splines, that is, a smooth curve network of the lowest possible computational complexity. Here, we show that the extremal interpolation problem for always has a solution. We give a characterization of this solution. We show that a solution to the problem for can be found by solving a system of nonlinear equations in the case where it exists.