  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.
