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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 ofthe second derivative for $1<p\leq\infty$. The problem for $p=2$ was set andsolved by Andersson et al. (1995). Vlachkova (2019) extended the results in(Andersson et al., 1995) and solved the problem for $1<p<\infty$. The minimumedge convex $L_p$-norm network for $1<p<\infty$ is obtained from the solutionto 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. Thecorresponding extremal problem for $p=\infty$ remained open. Here we show thatthe extremal interpolation problem for $p=\infty$ always has a solution. Wegive a characterization of this solution. We show that a solution to theproblem for $p=\infty$ can be found by solving a system of nonlinear equationsin the case where it exists.

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