Edge convex smooth interpolation curve networks with minimum $L_{\infty}$-norm of the second derivative

We consider the extremal problem of interpolation of convex scattered data in $\mathbb{R}^3$ by smooth edge convex curve networks with minimal $L_p$-norm of the second derivative for $1<p\leq\infty$. 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 $1<p<\infty$. The minimum edge convex $L_p$-norm network for $1<p<\infty$ is obtained from the solution to a system of nonlinear equations with coefficients determined by the data. The solution in the case $1<p<\infty$ is unique for strictly convex data. The corresponding extremal problem for $p=\infty$ remained open. Here we show that the extremal interpolation problem for $p=\infty$ always has a solution. We give a characterization of this solution. We show that a solution to the problem for $p=\infty$ can be found by solving a system of nonlinear equations in the case where it exists.