A Lagrangian program detecting the Weighted Fermat-Steiner-Frechet multitree for a Frechet $N-$multisimplex in the $N-$dimensional Euclidean Space

Abstract

In this paper, we introduce the Fermat-Steiner-Frechet problem for a given $\frac{N(N+1)}{2}-$tuple of positive real numbers determining the edge lengths of an $N-$simplex in $\mathbb{R}^{N},$ in order to study its solution called the "Fermat-Steiner-Frechet multitree," which consist of a union of Fermat-Steiner trees for all derived pairwise incongruent $N-$simplexes in the sense of Blumenthal, Herzog for $N=3$ and Dekster-Wilker for $N\ge 3.$ We obtain a method to determine the Fermat-Steiner Frechet multitree in $\mathbb{R}^{N}$ based on the theory of Lagrange multipliers, whose equality constraints depend on $N-1$ independent solutions of the inverse weighted Fermat problem for an $N-$simplex in $\mathbb{R}^{N}.$ A fundamental application of the Lagrangian program for the Fermat-Steiner Frechet problem in $\mathbb{R}^{N}$ is the detection of the Fermat-Steiner tree with global minimum length having $N-1$ equally weighted Fermat-Steiner points among $\frac{[\frac{N(N+1)}{2}]!}{(N+1)!}$ incongruent $N-$simplexes determined by an $\frac{N(N+1)}{2}-$tuple of consecutive natural numbers controlled by Dekster-Wilker, Blumenthal-Herzog conditions and enriched with the fundamental evolutionary processes of Nature (Minimum communication networks, minimum mass trasfer, maximum volume of incongruent simplexes). Furthermore, we obtain the unique solution of the inverse weighted Fermat problem, referring to the unique set of $(N+1)$ weights, which correspond to the vertices of an $N-$ boundary simplex in $\mathbb{R}^{N}.$