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.

翻译：我们考虑的是以1美元<p\leq\infty$每平端曲线网平滑端端端锥形曲线网以1美元=p$-norm来对流数据进行内插的极端问题。 美元=2美元的问题由Andersson等人(1995年)。 Vlachkova(2019年)扩大了(Andersson等人,1995年)的结果范围,并解决了1美元=p ⁇ infty$的问题。 1美元=p ⁇ p$-infty$的最小端端锥形锥体内插问题。 1美元=p ⁇ infty$的最小端端端锥形锥体内插问题总是有一个解决方案。 我们从数据确定的非线性方方方方方方程式系统中的解决方案中获取。 我们展示了1美元对等方程式的解决方案, 在那里可以找到, 方方程式的答案是 。