A Lagrangian Program Detecting the Weighted Fermat-Steiner-Fréchet Multitree for a Fréchet N-multisimplex in Euclidean N-space

We introduce the Fermat-Steiner-Fréchet (FSFR) problem for a given \(\frac{1}{2} N(N+1)\)-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 “FSFR multitree”, which consist of a union of Fermat-Steiner (FS) 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 FSFR 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 FSFR problem in \(\mathbb {R}^{N}\) is the detection of the FS tree with global minimum length having \(N-1\) equally weighted FS points among \(\frac{[\frac{1}{2} N(N+1)]!}{(N+1)!}\) incongruent N-simplexes determined by an \(\frac{1}{2} N(N+1)\)-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 transfer, 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 \(\mathbb {R}^{N}\). Additionally, we give a negative answer for an intermediate weighted Fermat-Steiner-Fréchet multitree having one node (weighted Fermat point) for m boundary closed polytopes (\(m\ge N+2\)), which is determined by m prescribed rays meeting at a fixed weighted Fermat point, by deriving a linear dependence for the m variable weights (Plasticity of an Intermediate FSFR multitree for m boundary closed polytopes). By entering in the plasticity of an intermediate FSFR multitree for m boundary closed polytopes a two-way mass transport from k vertices to the unique weighted Fermat point and from this point to the \(m-k\) vertices, and reversely, we derive the equations of “mutation” of intermediate FSFR multitrees in \(\mathbb {R}^{N}\).