# Edge convex smooth interpolation curve networks with minimum L_∞-norm of the second derivative

We consider the extremal problem of interpolation of convex scattered data in ℝ^3 by smooth edge convex curve networks with minimal L_p-norm of the second derivative for 1<p≤∞. The problem for p=2 was set and solved by Andersson et al. (1995). Vlachkova (2019) extended the results in (Andersson et al., 1995) and solved the problem for 1<p<∞. The minimum edge convex L_p-norm network for 1<p<∞ is obtained from the solution to a system of nonlinear equations with coefficients determined by the data. The solution in the case 1<p<∞ is unique for strictly convex data. The corresponding extremal problem for p=∞ remained open. Here we show that the extremal interpolation problem for p=∞ always has a solution. We give a characterization of this solution. We show that a solution to the problem for p=∞ can be found by solving a system of nonlinear equations in the case where it exists.

READ FULL TEXT