We consider the extremal problem of interpolation of convex scattered data

in R3 by smooth edge convex curve networks with minimal Lp-norm of the

second derivative for 1 < p ≤ ∞. The problem for p = 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 < ∞. The minimum edge convex

Lp-norm network for 1 < p < ∞ is obtained from the solution to a system

of nonlinear equations with coefficients determined by the data. The solution

in the case 1 < p < ∞ is unique for strictly convex data. The corresponding

extremal problem for p = ∞ remained open. The case p = ∞ 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 = ∞ always has a solution. We give a characterization

of this solution. We show that a solution to the problem for p = ∞ can be found by solving a system of nonlinear equations in the case where it exists.

Abstract. We consider the extremal problem of interpolation of scattered data in ℝ³ by smooth curve networks with minimal L_p-norm of the second derivative for 1 < p ≤ ∞. The problem for p = 2 was set and solved by Nielson [1]. Andersson et al. [2] gave a new proof of Nielson's result by using a different approach. Vlachkova [3] extended the results in [2] and solved the problem for 1 < p < ∞. The minimum L_p-norm network for 1 < p < ∞ is obtained from the solution to a system of nonlinear equations with coefficients determined by the data. The solution in the case 1 < p < ∞ is unique. We denote the corresponding minimum L_p-norm network by F_p. In the case where p = ∞ we establish the existence of a solution of the same type as in the case where 1 < p < ∞. This solution on each edge of the underlying triangulation is a quadratic spline function with at most one knot. We denote this solution by F_∞ and prove that the minimum L_p-norm networks F_p converge to the minimum L_∞-norm network F_∞ as p → ∞.

Edge convex smooth interpolation curve networks with minimum
𝐿∞-norm of the second derivative
Krassimira Vlachkova
https://doi.org/10.48550/arXiv.2212.11981

Abstract: We consider the extremal problem of interpolation of convex scattered data in ℝ3 by smooth edge convex curve networks with minimal 𝐿𝑝-norm of the second derivative for 1 < 𝑝 ≤ ∞. 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 < ∞. The minimum edge convex 𝐿𝑝-norm network for 1 < 𝑝 < ∞ is obtained from the solution to a system of nonlinear equations with coefficients determined by the data. The solution in the case 1 < 𝑝 < ∞ is unique for strictly convex data. The corresponding extremal problem for 𝑝 = ∞ remained open. Here we show that the extremal interpolation problem for 𝑝 = ∞ always has a solution. We give a characterization of this solution. We show that a solution to the problem for 𝑝 = ∞ can be found by solving a system of nonlinear equations in the case where it exists.