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

TL;DR

This paper addresses the open problem of extremal convex interpolation of scattered data in 3D with minimal $L_{\infty}$-norm of the second derivative, establishing existence, characterization, and solution methods.

Contribution

It proves the existence of solutions for the $p=\infty$ case and provides a characterization and solution approach via nonlinear equations.

Findings

Existence of solutions for the $p=\infty$ case is established.

Solutions can be characterized and computed through nonlinear equations.

The problem extends previous results for $1<p<\infty$ to the $p=\infty$ case.

Abstract

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.