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