Extremal interpolation of convex scattered data in ℝ3 by smooth edge convex minimum L∞‐norm networks: Characterization and solution.

We consider the extremal problem of interpolation of convex scattered data in ℝ3$$ {\mathrm{\mathbb{R}}}&#x0005E;3 $$ by smooth edge convex curve networks with minimal Lp$$ {L}_p $$‐norm of the second derivative for 1<p≤∞$$ 1<p\le \infty $$. The problem for p=2$$ p&#x0003D;2 $$ was set and solved by Andersson et al. (1995). Vlachkova (2019) extended the results of Andersson et al. (1995) and solved the problem for 1<p<∞$$ 1<p<\infty $$. The minimum edge convex Lp$$ {L}_p $$‐norm network for 1<p<∞$$ 1<p<\infty $$ is obtained from the solution to a system of nonlinear equations with coefficients determined by the data. The solution in the case 1<p<∞$$ 1<p<\infty $$ is unique for strictly convex data. The corresponding extremal problem for p=∞$$ p&#x0003D;\infty $$ remained open. The case p=∞$$ p&#x0003D;\infty $$ is of particular interest in the context of applications since it has a solution which is a smooth curve network consisting of quadratic splines, that is, a smooth curve network of the lowest possible computational complexity. Here, we show that the extremal interpolation problem for p=∞$$ p&#x0003D;\infty $$ always has a solution. We give a characterization of this solution. We show that a solution to the problem for p=∞$$ p&#x0003D;\infty $$ can be found by solving a system of nonlinear equations in the case where it exists. [ABSTRACT FROM AUTHOR]