# Isometric embedding of a weighted Fermat-Frechet multitree for isoperimetric
deformations of the boundary of a simplex to a Frechet multisimplex in the
$K$-Space

Anastasios N. Zachos  University of Patras, Department of Mathematics,
GR-26500 Rion, Greece<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study the weighted Fermat-Frechet problem for a
$\frac{N(N+1)}{2}-$tuple of positive real numbers determining $N$-simplexes in
the $N$ dimensional $K$-Space ($N$-dimensional Euclidean space
$\mathbb{R}^{N}$ if $K=0,$ the $N$-dimensional open hemisphere of radius
$\frac{1}{\sqrt{K}}$ ($\mathbb{S}_{\frac{1}{\sqrt{K}}}^{N}$) if $K>0$ and the
Lobachevsky space $\mathbb{H}_{K}^{N}$ of constant curvature $K$ if $K<0$).
The (weighted) Fermat-Frechet problem is a new generalization of the
(weighted) Fermat problem for $N$-simplexes.

We control the number of solutions (weighted Fermat trees) with respect to the
weighted Fermat-Frechet problem that we call a weighted Fermat-Frechet
multitree, by using some conditions for the edge lengths discovered by
Dekster-Wilker. In order to construct an isometric immersion of a weighted
Fermat-Frechet multitree in the $K$\- Space, we use the isometric immersion of
Godel-Schoenberg for $N$-simplexes in the $N$-sphere and the isometric
immersion of Gromov (up to an additive constant) for weighted Fermat (Steiner)
trees in the $N$-hyperbolic space $\mathbb{H}_{K}^{N}$.

Finally, we create a new variational method, which differs from Schafli’s,
Luo’s and Milnor’s techniques to differentiate the length of a geodesic arc
with respect to a variable geodesic arc, in the 3$K$-Space. By applying this
method, we eliminate one variable geodesic arc from a system of equations,
which give the weighted Fermat-Frechet solution for a sextuple of edge lengths
determining (Frechet) tetrahedra.

###### Key words and phrases:

weighted Fermat-Frechet problem, weighted Fermat-Frechet solution, weighted
minimum tree, spherical simplex, hyperbolic simplex, $K-$space, $N-$sphere,
$N-$hyperbolic space, isometric immersion

###### 2010 Mathematics Subject Classification:

Primary 51K05; Secondary , 52B12, 05C05, 51M09

##  1\. Introduction

In [20], Frechet posed the following problem in distance geometry (see also in
[41]):

Let $a_{ik}=a_{ki},i\neq k;i,k=0,1,2,\cdots,N$ ($a_{ik}=0,$ for $i=k$) be
$\frac{N(N+1)}{2}$ given positive quantities. What are the necessary and
sufficient conditions that they be the lengths of an $N$-simplex
$A_{0}A_{1}A_{2}\cdots A_{N}$ in $\mathbb{R}^{N}$ ?

Schoenberg restates and solves Frechet problem in a more general form ([41,
Theorem 1]):

A necessary and sufficient condition that the $a_{ik}$ be the lengths of the
edges of an $r$-simplex $A_{0}A_{1}A_{2}\cdots A_{N}$ lying in a Euclidean
space $\mathbb{R}^{r}$ ($1\leq r\leq N$) but not in a $\mathbb{R}^{r-1}$
($a_{ik}=0$ if and only if $i=k$) is

$F(x_{1},x_{2},...,x_{N})=\frac{1}{2}\sum_{i,k=1}^{N}(a_{0i}^{2}+a_{0k}^{2}-a_{ik}^{2})x_{i}x_{k}\geq
0$

and

$rank(F(x_{1},x_{2},...,x_{N}))=r.$

An extension of the Frechet problem for the construction of simplexes in the
$(r-1)$-dimensional spherical space $\mathbb{S}_{p}^{r-1}$ of radius $\rho$ is
given in [41, Theorem 2].

Necessary and sufficient conditions that the $a_{ik}$ be the
$\frac{N(N-1)}{2}$ lengths (mutual spherical distances) of the edges of an
$r-1$-simplex $A_{1}A_{2}\cdots A_{N}$ lying in $\mathbb{S}_{\rho}^{r-1}$
($1<r\leq N$) but not in a $\mathbb{S}_{\rho}^{r-2}$ for $i,k=1,2,\cdots,N$
are:

$\varphi(x_{1},x_{2},...,x_{N})=\frac{1}{2}\sum_{i,k=1}^{n}\cos(\frac{a_{ik}}{\rho})x_{i}x_{k}\geq
0,$

$a_{ik}\leq\pi\rho,$

$rank(\varphi(x_{1},x_{2},\cdots,x_{N}))=r.$

Menger independently solved the Euclidean Frechet problem obtaining equations
and inequalities by using determinants ([36, Third Fundamental theorem,pp.
738-743]) and Blumenthal-Garrett, Klanfer extends Menger’s method of
determinants for spherical simplexes in $S_{\rho}^{n}$ of radius $\rho$ ([4],
[34]).

A difficult problem is to determine $\frac{\frac{1}{2}N(N+1)!}{(N+1)!}$
pairwise incongruent $N$-simplexes with the same $\frac{1}{2}N(N+1)$ edges
$N>1$ in $\mathbb{R}^{N},$

In [5], [21], Blumenthal and Hertog, respectively, studied the Euclidean case
for $N=3$ and obtained conditions for six edges forming thirty incongruent
tetrahedra in $\mathbb{R}^{3}.$

Dekster and Wilker discovered necessary and sufficient conditions for
incongruent simplexes on spaces with constant curvature ([12], [14]), by
applying ideas and techniques taken from the theory of distance matrices first
developed by Seidel ([42]) and Neumaier ([34]). These conditions offer a
generalization of the triangle inequality for simplexes.

The $N$-dimensional $K$-space ($\mathbb{E}_{K}^{N}$). is the $N$-dimensional
Euclidean space $\mathbb{R}^{N}$ if $K=0,$ the $N$-dimensional open hemisphere
of radius $\frac{1}{\sqrt{K}}$ ($\mathbb{S}_{\frac{1}{\sqrt{K}}}^{N}$) if
$K>0$ and the Lobachevsky space $\mathbb{H}_{K}^{N}$ of constant curvature $K$
if $K<0.$

The weighted Fermat problem in the $\mathbb{E}_{K}^{N}$ refers to finding the
unique point minimizing the sum of geodesic distances from this point to each
point from a finite set of fixed non-collinear points multiplied by a positive
real number (weight), which corresponds to each distance. The solution is
referred as the weighted Fermat point. If the weighted Fermat point does not
belong to this finite set of fixed points (vertices), the solution is called a
weighted (floating) Fermat-Torricelli point, otherwise it is called a weighted
(absorbing) Fermat-Torricelli point. The branching solution, which consists of
the weighted branches that connect the weighted Fermat point with each fixed
vertex is called a weighted Fermat tree.

We focus on the connection of the weighted Fermat problem with the Frechet
problem (weighted Fermat-Frechet problem) in the $\mathbb{E}_{K}^{N}.$

######  Problem 1 (Weighted Fermat-Frechet problem).

Find the discrete set of weighted Fermat trees, which correspond to a given
$N-$tuple of positive real numbers determining $\frac{1}{2}N(N+1)$ edges of
incongruent $N$-simplexes in $\mathbb{E}_{K}^{N}.$

In [5], Blumenthal claimed that the maximum number of incongruent
$(N+1)-$simplexes in $\mathbb{R}^{N}$, which may be obtained by a given
$N-$tuple of positive real numbers determining $\frac{1}{2}N(N+1)$ edges is
$\frac{\frac{1}{2}N(N+1)!}{(N+1)!}.$

In [50] and [51], we studied the weighted Fermat-Frechet problem for $N=3$
(geodesic triangles) in the two dimensional $K$-plane ($\mathbb{E}_{K}^{2}$)
and we found the position of the weighted Fermat-Torricelli trees, which
yields the weighted Fermat-Frechet (multitree) solution.

In [53], we found the position of the weighted Fermat-Torricelli trees for the
weighted Fermat-Frechet problem for a given sextuple of positive real numbers
determining the edge lengths of tetrahedra in $\mathbb{R}^{3},$ by
substituting Caley-Menger determinants in some weighted volume entropy
equalities for tetrahedra derived in [49].

In the present paper, we introduce a class of weighted Fermat-Torricelli trees
(Fermat-Frechet multitree) for a given $\frac{N(N+1)}{2}$-tuple of positive
real numbers determining $N$-simplexes in $\mathbb{E}_{K}^{N}.$ We apply ideas
and methods by Godel, Schoenberg and Gromov, in order to construct isometric
immersions of a weighted Fermat-Frechet multitree for isoperimetric
deformations of the boundary of incongruent simplexes in simplexes inside
$\mathbb{E}_{K}^{N}.$

The main results of the paper are the following:

First, we study the conditions for the solution of the weighted Fermat-Frechet
problem (Fermat-Frechet multitree), which corresponds to a union of weighted
Fermat-Torricelli trees for all incongruent $N$-simplexes. The family of
incongruent $N$-simplexes, whose edge lengths satisfy the conditions given by
Dekster-Wilker ([12],[13],[14],[15]) form a Frechet multisimplex in
$\mathbb{E}_{K}^{N}$ (Theorems 1, 3, 6, 8).

Next we use the ideas of Godel-Schoenberg ([41]), which focus on Godel’s
observation of thinking of the edges of an $N$-(Euclidean or spherical)simplex
to made of fexible strings and on placing in the interior a small sphere,
which was gradually inflated. When the sphere reaches a definite size, it will
be tightly packed within the edges (strings) of the $N$-simplex. Our idea is
to consider the case of tightly packed a weighted Fermat-Torricelli tree with
respect to a boundary $(N-1)$-simplex in a proper $(N-1)$ dimensional sphere.
Thus, following Godel-Shoenberg methods reversely we derive some controlled
isometric immersions of a weighted Fermat-Frechet multitree to a Frechet
simplex in a larger spherical space (Theorems 11, 14). The conditions of
Dekster-Wilker need to be satisfied, in order to control the isometric
immersions of Frechet multisimplex enriched with the corresponding weighted
Fermat trees, which contains all incongruent simplexes.

We apply the theory of Gromov’s isometric embedding of geodesic trees in the
$N$ dimensional hyperbolic space of constant curvature $K$
$\mathbb{H}_{K}^{N}$ ([24]), in order to construct an embedding (inclusion
map) of a Fermat-Steiner-Frechet multitree solution for a given
$\frac{N(N+1}{2}$-tuple of edge lengths determining incongruent boundary
$N$-simplexes to an associated family of Gromov isometries up to an additive
constant for $N$-simplexes and ideal $N$-simplexes in $\mathbb{H}_{K}^{N}.$
The Fermat-Steiner-Frechet multitree is a union of (intermediate) weighted
Fermat-Steiner trees having upto $N-2$ nodes (Fermat-Steiner points) inside
the $N$-simplex and may be considered as a generalization of weighted minimal
binary trees studied by Ivanov and Tuzhilin in [28] (Theorems 15, 16).
Furthermore, we apply Gromov’s theory of $\delta$ thin metric trees in
hyperbolic spaces, in order to measure the reduction of intelligence of
intermediate Fermat-Steiner trees (Number of Fermat Steiner nodes), by
applying Gromov’s convergence of ideal simplexes to two subsimplexes (Theorem
17).

We continue, by giving a new variational method, which focus on the
generalization of cosine law in the 3-$K$-Space (Theorem 18). We note that a
generalization of the cosine law in $\mathbb{R}^{3}$ has been given in [49].
This variational method differs from Schafli ([40]), Luo’s ([35])) and
Milnor’s ([37]) techniques to differentiate the length of a geodesic arc with
respect to a variable geodesic arc in the 3-$K-$Space. By applying this method
to a system of equations that deal with the weighted Fermat-Frechet solution
in $\mathbb{E}_{K}^{N},$ for $N=4,$ we eliminate one variable geodesic arc
from the equations and we determine the weighted Fermat-Frechet solution for a
sextuple of edge lengths for Frechet multitetrahedra (Theorem 19).

Finally, we conclude with some calculations for the determination of weighted
Fermat trees for tetrahedra having one or three vertices at infinity in
$\mathbb{R}^{3}$ (Theorem 21).

The paper is organized as follows: In Section 2, we study the conditions to
solve the weighted Fermat-Frechet problem in the $K-$Space (Theorems 1- 9,
Proposition 1, 2, 3)

In Section 3, we construct a controlled Godel-Schoenberg isometric immersion
of the weighted Fermat Frechet multitree for $N-1$ boundary Frechet
multisimplex in $\mathbb{S}_{\rho_{1}}^{N-2}$ to a class of weighted simplexes
in $\mathbb{S}_{\rho_{0}}^{N-1}$ for $\rho_{0}>\rho_{1}$ (Theorems 9-14).

In Section 4, we construct a Gromov isometric immersion up to an additive
constant for weighted Fermat-Steiner Frechet multitrees in
$\mathbb{H}_{K}^{N}$ (Theorem 15- 17)

In Section 5,we give a new variational method to derive the weighted Fermat-
Frechet multitree for Frechet multitetrahedra in the $3-K-$Space (Theorem
18-19, Proposition 4).

In the last section, we present a computational method to obtain weighted
Fermat-Torricelli trees with respect to a tetrahedron having one or three
(ideal) vertices at infinity (Theorem 20-21, Propositions 7-8).

##  2\. The weighted Fermat-Frechet problem in the $N$ dimensional $K$-space

In this section, we introduce the weighted Fermat-Frechet problem for a given
$\frac{N(N+1)}{2}$-tuple of positive real numbers determining $N$-simplexes in
$\mathbb{E}_{K}^{N}$ and we study the conditions to obtain a class of weighted
Fermat-Torricelli trees, which form the weighted Fermat-Frechet multitree
solution.

We start by describing the Dekster-Wilker domain given in [12], [13],
[14],[15], which gives the maximum number of mutually incongruent simplexes
for the same $\frac{N(N+1)}{2}$-tuple of edges in $\mathbb{E}_{K}^{N}.$

We denote by $\\{A_{1},A_{2},\cdots,A_{N}\\}$ an $(N-1)$-simplex, by $A_{0}$
an interior point and by
$\\{A_{1},A_{2},\cdots,A_{0}\cdots,A_{N}\\}\subset\\{A_{1},A_{2},\cdots,A_{N}\\}$
an $(N-1)$-simplex, which is derived by replacing the vertex
$A_{j}\in\\{A_{1},A_{2},\cdots,A_{N}\\}$ with $A_{0}$ in $\mathbb{E}_{K}^{N}.$

A. The weighted Fermat-Frechet problem in $\mathbb{R}^{N}$

Dekster and Wilker use the notations $\ell=\max_{i,j}a_{ij},$
$s=\min_{i,j}a_{ij}$ and

$\lambda_{N}(\ell)=\left\\{\begin{array}[]{ll}\ell\sqrt{1-\frac{2(N+1)}{N(N+2)}}&for\
even\ N\geq 2,\\\ \ell\sqrt{1-\frac{2}{(N+1)}}&for\ odd\ N\geq
3\par\end{array}\right.$

######  Definition 1.

The Dekster-Wilker Euclidean domain $DW_{\mathbb{R}^{N}}(\ell,s)$ is a closed
domain in $\mathbb{R}^{2}$ between the ray $s=\ell,$ and the graph of a
function $\lambda_{N}(\ell),$ $\ell\geq 0,$ which is less than $\ell$ for
$\ell\neq 0,$

######  Lemma 1.

Dekster-Wilker incongruent simplexes in $\mathbb{R}^{N},$ [13, (1.3),(1.4)]

If $s\leq a_{ij}\leq\ell,$ for $\ell,s\in DW_{\mathbb{R}^{N}}(\ell,s),$ then
$a_{ij}$ determine the edge lengths of incongruent simplexes in
$\mathbb{R}^{N}.$

The characterization of solutions (absorbing/floating Fermat-Torricelli trees)
of the Fermat-problem in $\mathbb{R}^{N}$ was first given by Sturm in [44] and
it was extended for the weighted case by Kupitz and Martini ([6, Chapter II,
Theorem 18.37]) using calculus. These two cases of weighted Fermat trees are
used to derive the weighted Fermat-Frechet solution for a given
$\frac{N(N+1)}{2}-$tuple of positive real numbers $a_{ij}$ determining the
edge lengths of incongruent $N$-simplexes in $\mathbb{R}^{N}.$

######  Theorem 1 (The weighted Fermat-Frechet solution for $N$-simplexes in
$\mathbb{R}^{N}$).

The weighted Fermat-Frechet solution for a given $\frac{N(N+1)}{2}$-tuple
$a_{ij},$ such that $s\leq a_{ij}\leq\ell,$ for $\ell,s\in
DW_{\mathbb{R}^{N}}(\ell,s)$ consists of a maximum number of
$\frac{\frac{1}{2}N(N+1)!}{(N+1)!}$ weighted Fermat trees, which belong to one
of the following two cases:

(I) If for each index $k\in\\{1,2,3,\cdots,N\\},$

$\sum_{i=1,i<j}^{N}B_{i}B_{j}\frac{a_{ik}^{2}+a_{jk}^{2}-a_{ij}^{2}}{2a_{ik}a_{jk}}>\frac{B_{k}^{2}-\sum_{i=1,i<j}^{N}B_{i}^{2}}{2},$
(2.1)

we obtain the weighted floating Fermat-Torricelli tree
$\\{a_{01},a_{02},\cdots,a_{0N}\\}.$

(II) If there is an index $k\in{1,2,\cdots,N},$ such that:

$\sum_{i=1,i<j}^{N}B_{i}B_{j}\frac{a_{ik}^{2}+a_{jk}^{2}-a_{ij}^{2}}{2a_{ik}a_{jk}}\leq\frac{B_{k}^{2}-\sum_{i=1,i<j}^{N}B_{i}^{2}}{2},$
(2.2)

we obtain the weighted absorbing Fermat-Torricelli tree
$\\{a_{k1},a_{k2},\cdots,a_{kN}\\}.$

###### Proof.

The edge lengths $a_{ij}$ yield a simplex $A_{1}A_{2}\cdots A_{N+1}$ in
$\mathbb{R}^{N}.$ Let $X$ be a point in $\mathbb{R}^{N}.$ We assume that a
positive number $B_{i}$ (weight) correspond to each length of the segment
$[X,A_{i}],$ for $i=1,2,\cdots,N.$

By applying the method of directional derivatives by Kupitz and Martini [6,
Chapter II, Theorem 18.16] for the weighted case and by applying the cosine
law in $\triangle A_{i}A_{j}A_{k},$ we get (2.1) and (2.2).

The existence and uniqueness of the weighted Fermat-Torricelli point $X\equiv
A_{0}$ proved by convexity and compactness arguments gives the existence and
uniqueness of the weighted Fermat-Torricelli trees for each simplex, which
belongs to the class of incongruent boundary simplexes. Thus, the maximum
number of arrengements of $\frac{N(N+1)}{2}$ edges gives
$\frac{\frac{1}{2}N(N+1)!}{(N+1)!}$ mutually incongruent simplexes, which
correspond to $\frac{\frac{1}{2}N(N+1)!}{(N+1)!}$ weighted floating and
absorbing Fermat-Torricelli trees.

∎

######  Theorem 2 (The weighted Euclidean Fermat-Frechet problem for large
$N$).

The weighted Fermat-Frechet solution for a given large number $N$ and
$\frac{N(N+1)}{2}-$tuple of edge lengths determining simplexes, consists of a
maximum number of
$\frac{(\frac{N(N+1)}{2})^{\frac{N(N+1)+1}{2}}}{(N+1)^{N+\frac{3}{2}}}\exp(-(N+1)(\frac{N}{2}-1)+\frac{\theta_{1}}{6N(N+1)}-\frac{\theta_{2}}{12(N+1)})$
weighted Fermat trees, for $\theta_{1},\theta_{2}\in(0,1).$

###### Proof.

It is a direct consequence of Theorem 1 by using Stirling’s formula ([1,
6.1.38, p. 257]):

$x!=\sqrt{2\pi}x^{x+\frac{1}{2}\exp(-x+\frac{\theta}{12x})},$ (2.3)

for $x>0$ and $\theta\in[0,1].$

By substituting in $\ref{Stirling1},$ $x\equiv\frac{N(N+1)}{2}$ and $x\equiv
N+1$ for $\theta_{1},\theta_{2}\in(0,1),$ and dividing the two derived
relations gives an upper bound of
$\frac{(\frac{N(N+1)}{2})^{\frac{N(N+1)+1}{2}}}{(N+1)^{N+\frac{3}{2}}}\exp(-(N+1)(\frac{N}{2}-1)+\frac{\theta_{1}}{6N(N+1)}-\frac{\theta_{2}}{12(N+1)})$
weighted Fermat trees that correspond to the mutually incogruent simplexes in
$\mathbb{R}^{N}.$

∎

The volume of an $N$-simplex $A_{1}A_{2}\cdots A_{N}$ in $\mathbb{R}^{N}$ is
given by the Caley-Menger determinant in terms of edge lengths ([43, (5.1), p.
125]):

$\operatorname{Vol}(A_{1}A_{2}\cdots A_{N})^{2}=\\\
\frac{1}{(-1)^{N}2^{N-1}((N-1)!)^{2}}\begin{vmatrix}0&1&1&\cdots&1&1\\\
1&0&a_{12}^{2}&\cdots&a_{1(N-1)}^{2}&a_{1N}^{2}\\\
1&a_{21}^{2}&0&\cdots&a_{2(N-1)}^{2}&a_{2N}^{2}\\\ .&.&.&.&.&.\\\
1&a_{N1}^{2}&a_{N2}^{2}&\cdots&a_{N(N-1)}^{2}&0\end{vmatrix}.$

######  Theorem 3 (The Euclidean Fermat-Torricelli Frechet solution).

The following equations depending on the variable lengths
$a_{01},a_{02},\cdots,a_{0N}$ and the segments $\\{a_{ij}\\}$ provide a
necessary condition for the determination of the weighted floating Fermat-
Torricelli trees $\\{a_{01},a_{02},\cdots,a_{0N}\\},$ which belong to the
weighted Fermat-Frechet solution:

$\sum_{i=2}^{N}\frac{B_{i}}{a_{0i}}(a_{01}^{2}+a_{0i}^{2}-a_{1i}^{2})-\sum_{i=1,{i\neq
j}}^{N}\frac{B_{i}}{a_{0i}}(a_{0j}^{2}+a_{0i}^{2}-a_{ji}^{2})=-2(B_{1}a_{01}-B_{j}a_{0j}),$
(2.4)
$\operatorname{Vol}(A_{1},A_{2},\cdots,A_{N})=\sum\operatorname{Vol}(A_{1},A_{2},\cdots,A_{0},\cdots,A_{N}),$
(2.5)

for $j=2,\cdots,N.$

###### Proof.

It is well known that the first variational formula of a line segment
$A_{0}A_{i}$ with respect to a physical parameter $A_{0}A_{j}$ (line segment),
which meet at a point $A_{0}$ in $\mathbb{R}^{N}$ is given by:

$\frac{da_{0i}}{da_{0j}}=\cos\angle A_{i}A_{0}A_{j},$ (2.6)

for $i,j=1,2,,\cdots,N.$ Differentiating the objective function
$f(a_{01},a_{02},\cdots,a_{0N})=\sum_{i=1}^{N}B_{i}a_{0i}$ with respect to
$a_{0i},$ for $i=1,2,\cdots N,$ we get:

$\sum_{i=1}^{N}B_{i}\cos(\alpha_{10i})=0,$ (2.7)
$\sum_{i=1}^{N}B_{i}\cos(\alpha_{20i})=0,$ (2.8)

$\cdots$

$\sum_{i=1}^{N}B_{i}\cos(\alpha_{N0i})=0.$ (2.9)

By subtracting (2.8)-(2.9) from (2.7) and by applying the cosine law in
$\triangle A_{j}A_{0}A_{i}$ for $j=1,2,\cdots,N,$ we derive (2.4).

The interior point $A_{0}$ of $A_{1}A_{2}\cdots A_{N}=\cup{A_{1}A_{2}\cdots
A_{0}\cdots A_{n}}$ gives the property of addition of Euclidean volumes of
simplexes in $\mathbb{R}^{N}.$ By substituting in
$\operatorname{Vol}(A_{1},A_{2},\cdots,A_{N}),$
$\operatorname{Vol}(A_{1},A_{2},\cdots,A_{0},\cdots,A_{N})$ the corresponding
Caley Menger determinant depending on the edge lengths $a_{ij},$ for
$i,j=0,1,2,\cdots,N,$ we obtain (2.5).

∎

The edge lengths $a_{0i}$ are the branches of the weighted Fermat tree
$\\{a_{01},a_{02},\cdots,a_{0N}\\}$ consist the branching solution of the
weighted Fermat problem. The field of branching solutions in geometric
optimization has been developed by Ivanov and Tuzhilin in [30].

######  Lemma 2 (Isoperimetric inequality of a regular simplex in
$\mathbb{R}^{N}$).

[27, p. 274] An Euclidean $N$-simplex is of maximal volume if it is regular.

######  Proposition 1.

Each branch of the weighted Fermat-Frechet solution is less than the diameter
$2\sqrt{\frac{N}{a(N+1)}}$ of the circumscribed sphere of a regular simplex
with equal edge length $a\equiv\frac{\sum_{i,j}a_{ij}}{\frac{N(N+1)}{2}}.$

###### Proof.

The radius of the circumsphere of a regular $N$ simplex in $\mathbb{R}^{N}$ is
given by $2\sqrt{\frac{N}{a(N+1)}.}$ By applying lemma 2, we derive that:

$a_{0i}<2\sqrt{\frac{N}{a(N+1)}},$

for $a\equiv\frac{\sum_{i,j}a_{ij}}{\frac{N(N+1)}{2}}.$ ∎

B. The weighted Fermat-Frechet problem in the hyperbolic space
$\mathbb{H}_{K}^{N},$ for $K<0.$

Dekster and Wilker ([13, (1.5),(1.6)],[15]) describe a class of incongruent
hyperbolic simplexes in terms of $\ell,s,$ such that: $\ell=\max_{i,j}a_{ij},$
$s=\min_{i,j}a_{ij}$ and

$\lambda_{N}(\ell)=$
$\left\\{\begin{array}[]{ll}\frac{1}{K}\cosh^{-1}(\sqrt{1+2(1-\frac{2}{N})\sinh^{2}K\frac{\ell}{2}}\sqrt{1+2(1-\frac{2}{N+2})\sinh^{2}K\frac{\ell}{2}})&for\
even\ N\geq 2,\\\ \frac{2}{K}\sinh^{-1}(\sqrt{1-\frac{2}{(N+1)}}\sinh
K\frac{\ell}{2})&for\ odd\ N\geq 3\end{array}\right.$

######  Definition 2.

The Dekster-Wilker hyperbolic domain $DW_{\mathbb{H}_{K}^{N}}(K\ell,Ks)$ is a
closed domain in $\mathbb{R}^{2}$ between the ray $s=\ell,$ and the graph of a
function $\lambda_{N}(\ell),$ $\ell\geq 0,$ which is less than $K\ell$ for
$K\ell\neq 0,$

######  Lemma 3.

Dekster-Wilker incongruent simplexes in $\mathbb{H}^{N},$ [13, (1.5),(1.6)]

If $s\leq a_{ij}\leq\ell,$ for $K\ell,Ks\in
DW_{\mathbb{H}_{K}^{N}}(K\ell,Ks),$ then $a_{ij}$ determine the edge lengths
of incongruent simplexes in $\mathbb{H}_{K}^{N}.$

In [39, Theorem 1, Proposition 2,p. 97], Noda, Sakai, Morimoto derive a
characterization of solutions (absorbing/floating Fermat-Torricelli trees)for
the Fermat-problem in $\mathbb{H}_{K}^{N}$ and more general for simply
connected smooth Riemannian manifolds with non-positive sectional curvature
(Hadamard manifolds). An extension of characterizations of solutions of Fermat
trees for the weighted case leads to the weighted Fermat-Frechet solution for
a given $\frac{N(N+1)}{2}$-tuple of positive real numbers $a_{ij}$ determining
the edge lengths of incongruent $N$-simplexes in $\mathbb{H}_{K}^{N}.$

We cannot reformulate Theorem 3 for the hyperbolic case, by using directly the
addition property of Volumes of hyperbolic simplexes. In [37, Theorem, p.
200], Milnor shows that the addition property of the Lobachevsky function,
which is connected with hyperbolic volume holds for ideal (having their
vertices at infinity) hyperbolic 3-simplexes and computes by using power
series the volume of an ideal regular $N$-simplex [37, pp. 206-207].

In [26], Haagerup and Munkholm proved an isoperimetric inequality of
hyperbolic simplexes in $\mathbb{H}^{N}.$

######  Lemma 4.

Isoperimetric inequality of a hyperbolic regular simplex in
$\mathbb{H}_{K}^{N},$ [26] A hyperbolic $N$-simplex is of maximal volume if it
is ideal and regular.

######  Theorem 4 (The weighted Fermat-Frechet problem in
$\mathbb{H}_{K}^{N}$).

The weighted Fermat-Frechet solution for a given $\frac{N(N+1)}{2}-$tuple
$a_{ij},$ such that $s\leq a_{ij}\leq\ell,$ for $\ell,s\in
DW_{\mathbb{H}_{K}^{N}}(\ell,s)$ consists of a maximum number of
$\frac{\frac{1}{2}N(N+1)!}{(N+1)!}$ weighted Fermat trees, which belong to one
of the following two cases:

(I) If for each index $k\in\\{1,2,3,\cdots,N\\},$

$\sum_{i=1,i<j}^{N}B_{i}B_{j}\frac{\cosh Ka_{ik}\cosh Ka_{jk}-\cosh
Ka_{ij}}{\sinh Ka_{ik}\sinh
Ka_{jk}}>\frac{B_{k}^{2}-\sum_{i=1,i<j}^{N}B_{i}^{2}}{2},$ (2.10)

we obtain the weighted floating Fermat-Torricelli tree
$\\{a_{01},a_{02},\cdots,a_{0N}\\}.$

(II) If there is an index $k\in{1,2,\cdots,N},$ such that:

$\sum_{i=1,i<j}^{N}B_{i}B_{j}\frac{\cosh Ka_{ik}\cosh Ka_{jk}-\cosh
Ka_{ij}}{\sinh Ka_{ik}\sinh
Ka_{jk}}\leq\frac{B_{k}^{2}-\sum_{i=1,i<j}^{N}B_{i}^{2}}{2},$ (2.11)

we obtain the weighted absorbing Fermat-Torricelli tree
$\\{a_{k1},a_{k2},\cdots,a_{kN}\\}.$

###### Proof.

The existence and uniqueness of the weighted Fermat-Torricelli point $A_{0}$
in $\mathbb{H}_{K}^{N}$ is given by compactness and convexity arguments of the
distance function $a_{0i}$ following Thurston [46, Theorem 2.5.8] under the
weighted conditions of Noda-Sakai-Morimoto gives a maximum number of weighted
Fermat trees for $s\leq a_{ij}\leq\ell:$ $(Ks,K\ell)\in
DW_{\mathbb{H}_{K}^{N}}(\ell,s).$ By substituting the hyperbolic law of
cosines in $\triangle A_{i}AjA_{k}$ in the weighted norm inequalities of unit
tangent vectors, we obtain (2.10) and (2.11). ∎

######  Theorem 5 (The hyperbolic Fermat-Torricelli Frechet solution).

The following equations depending on the variable hyperbolic edge lengths
$a_{01},a_{02},\cdots,a_{0N}$ and the hyperbolic edge lengths $\\{a_{ij}\\}$
provide a necessary condition for the location of the weighted floating
Fermat-Torricelli trees $\\{a_{01},a_{02},\cdots,a_{0N}\\},$ which belong to
the weighted Fermat-Frechet multitree solution:

$\displaystyle\sum_{i=2}^{N}\frac{B_{i}}{\sinh Ka_{01}\sinh Ka_{0i}}(\cosh
Ka_{01}\cosh Ka_{0i}-\cosh Ka_{1i})-$ $\displaystyle\sum_{i=1,{i\neq
j}}^{N}\frac{B_{i}}{\sinh Ka_{0j}\sinh Ka_{0i}}(\cosh Ka_{0j}\cosh
Ka_{0i}-\cosh Ka_{ji})=0,$ (2.12)

$\operatorname{Vol(A_{1},A_{2},\cdots,A_{0}\cdots,A_{N})}<\sqrt{N}\sum_{k=0}^{\infty}\frac{\beta(\beta+1)\cdots(\beta+k-1)}{(N+2k)!}A_{N,k}$
(2.13)

with $\beta=\frac{N+1}{2},$ $A_{N,k}=\sum_{i_{0}+\cdots+i_{N}=k,i_{a}\geq
0}\frac{(2i_{0})!}{i_{0}!}\cdots\frac{(2i_{N})!}{i_{N}!},$ for $j=2,\cdots,N.$

###### Proof.

By differentiating the objective function $f(a_{01},a_{02},\cdots
a_{0N})=\sum_{i=1}^{N}B_{i}$ with respect to arc length and by using the first
variation formula of a variable arc length $a_{0i}$ with respect to arc length
$a_{01}$, and with respect to $a_{0j}$ and by subtracting the two derived
relations and then substituting $\cos\angle A_{i}A_{0}A_{j}$ by the hyperbolic
law of cosines in $\triangle A_{i}A_{0}A_{j},$ we get (5). By substituting in
Lemma 4 Milnor’s computation of a hyperbolic regular $N$ simplex (upper
bound), we derive (2.13).

∎

Assume that the Frechet class of hyperbolic simplexes contains an orthosimplex
$A_{N+1}^{\circ}A_{1}^{\circ}A_{2}^{\circ}\cdots A_{N}^{\circ}$ with edge
lengths $a_{ij}^{\circ}:$ $s\leq a_{ij}^{\circ}\leq\ell:$ $(Ks,K\ell)\in
DW_{\mathbb{H}_{K}^{N}}(K\ell,Ks).$ An $N$ dimensional orthosimplex (Euclidean
or non-Euclidean) is an $N-$simplex whose faces $F_{0},\cdots F_{N}$ satisfy:
$F_{i}\perp F_{j}$ for $|i-j|\geq 2$ ([37, p. 207]). By using Milnor’s elegant
computation of volume of a hyperbolic orthosimplex, we get a reformulation of
Theorem 6 using the following lemma:

######  Lemma 5.

Computation of the Volume of an orthosimplex in $\mathbb{H}_{K}^{N}$,[37, p.
208]

$\displaystyle\operatorname{Vol}((A_{N+1}^{\circ}A_{1}^{\circ},A_{2}^{\circ},\cdots,A_{N}^{\circ})=$
$\displaystyle a_{1}a_{2}\cdots a_{N}\sum_{i_{1},i_{2},\cdots
i_{N}=0}^{\infty}\frac{\beta(\beta+1)\cdots(\beta+k-1)a_{1}^{2i_{1}}\cdots
a_{N}^{2i_{N}}}{i_{1}!i_{2}!\cdots
i_{N}!(2i_{1}+\cdots+2i_{N}+N)\cdots(2i_{N}+1)}$ (2.14)

where

$k=i_{1}+i_{2}+\cdots+i_{N},$ $\beta=\frac{N+1}{2}$

$\tanh Ka_{(N+1)1}=a_{1},\tanh
Ka_{12}=\frac{a_{2}}{\sqrt{1-a_{1}^{2}}},\cdots,$
$\tanh Ka_{(N-1)N}=\frac{a_{N}}{\sqrt{1-a_{1}^{2}-\cdots-a_{N-1}^{2}}}.$

######  Theorem 6 (The hyperbolic Fermat-Torricelli Frechet solution).

The following equations depending on the variable hyperbolic edge lengths
$a_{01},a_{02},\cdots,a_{0N}$ and the hyperbolic edge lengths $\\{a_{ij}\\}$
provide a necessary condition for the location of the weighted floating
Fermat-Torricelli trees $\\{a_{01},a_{02},\cdots,a_{0N}\\},$ which belong to
the weighted Fermat-Frechet (orthosimplex) solution:

$\displaystyle\sum_{i=2}^{N}\frac{B_{i}}{\sinh Ka_{01}\sinh Ka_{0i}}(\cosh
Ka_{01}\cosh Ka_{0i}-\cosh Ka_{1i})-$ $\displaystyle\sum_{i=1,{i\neq
j}}^{N}\frac{B_{i}}{\sinh Ka_{0j}\sinh Ka_{0i}}(\cosh Ka_{0j}\cosh
Ka_{0i}-\cosh Ka_{ji})=0,$

$\operatorname{Vol}(A_{1},A_{2},\cdots,A_{0}\cdots,A_{N},A_{N+1})<\lambda\operatorname{Vol}(A_{N+1}^{\circ}A_{1}^{\circ},A_{2}^{\circ},\cdots
A_{N}^{\circ})+m.$ (2.15)

for some $\lambda,m>0.$

Given $\\{a_{01},a_{02},\cdots,a_{0N}\\}$ a weighted (floating) Fermat-
Torricelli tree with respect to the hyperbolic simplex $A_{1}A_{2}\cdots
A_{N}$ in $\mathbb{H}_{K}^{N}$ with unknown constant curvature $K,$ which
belongs to the hyperbolic weighted Fermat-Frechet multitree solution, we may
compute the curvature $K$ and the radius $R$ of the sphere of the Klein model,
by using the hyperbolic Caley-Menger determinant. The hyperbolic Caley Menger
determinant for $\\{A_{0},A_{1},\cdots,A_{N}\\}$ is given by:

$\det(A_{0},A_{1},A_{2},\cdots,A_{N})=\\\ \begin{vmatrix}1&\cosh Ka_{01}&\cosh
Ka_{02}&\cdots&\cosh Ka_{0N}\\\ \cosh Ka_{10}&1&\cosh Ka_{12}&\cdots&\cosh
Ka_{1N}\\\ \cosh Ka_{20}&\cosh Ka_{21}&1&\cdots&\cosh Ka_{2N}\\\
.&.&.&.&.&.\\\ \cosh Ka_{N0}&\cosh Ka_{N1}&\cosh
Ka_{N2}&\cdots&1\end{vmatrix}.$

Thus, by setting $\det(A_{0},A_{1},A_{2},\cdots,A_{N})=0,$ we may derive an
estimate of $K.$ In [45], Terence Tao used a spherical Caley-Menger
determinant to obtain estimate of the radius of the Earth.

######  Proposition 2.

If a weighted Fermat-Frechet multitree $\\{a_{01},a_{02},\cdots,a_{0N}\\}$ is
given in $\mathbb{H}_{K}^{N}$ then we get the same sphere on the Klein model.

###### Proof.

Solving $\det(A_{0},A_{1},A_{2},\cdots,A_{N})=0,$ for each weighted Fermat-
Torricelli tree that consists the weighted Fermat-Torricelli multitree yields
an estimate for the constant negative curvature $K$ and the corresponding
radius $R$ in the Klein model. ∎

C. The weighted Fermat-Frechet problem in the spherical space
$\mathbb{S}_{K}^{N},$ for $K>0$ and $R=\frac{1}{\sqrt{K}}.$

Dekster and Wilker ([13, (3.1),(3.6),(1.3),(1.5)],[15, pp. 9–11]) introduced a
class of incongruent spherical simplexes in terms of $\ell,s,$ such that:
$\ell=\max_{i,j}a_{ij},$ $s=\min_{i,j}a_{ij}$ and the following notations are
used:

$\lambda_{N}(\ell)=$
$\left\\{\begin{array}[]{ll}\frac{1}{K}\arccos(\sqrt{1+2(1-\frac{2}{N})\sin^{2}K\frac{\ell}{2}}\sqrt{1+2(1-\frac{2}{N+2})\sin^{2}K\frac{\ell}{2}})&for\
even\ N\geq 2,\\\ \frac{2}{K}\arcsin(\sqrt{1-\frac{2}{(N+1)}}\sin
K\frac{\ell}{2})&for\ odd\ N\geq 3\end{array}\right.,$

$m_{N}(K\ell)=\frac{2}{K}\arcsin K\sqrt{\frac{2+2(N-1)\cos K\ell}{1+(N-2)\cos
K\ell}}\sin K\frac{\ell}{2}$

$\ell_{N}^{\star}=\frac{2}{K}\arcsin K\sqrt{\frac{N+1}{2N}},$

$a_{N}=\frac{2}{K}\arcsin\frac{K}{4}\frac{\sqrt{7N-4+\sqrt{N^{2}+8N}}}{N-1},$

$K\ OC:K\ s=K\ell,K\ell\in[0,K\ell_{N}^{\star}],$

$K\ CF:K\ s=K\ m_{N}(K\ell),K\ell\in[K\ell_{N}^{\star},Ka_{N}],$

$K\ OE:K\ s=K\lambda_{N}(K\ \ell),K\ell\in[0,K\ell_{N}^{\star}],$

$EF:Ks=K\lambda_{N}(K\ell),K\ \ell\in[K\ell_{N}^{\star},Ka_{N}].$

######  Definition 3.

[13],[15] The Dekster-Wilker spherical domain
$(K\ell,Ks)\in DW_{\mathbb{S}_{K}^{N}}(K\ell,Ks)$ is a closed domain in
$\mathbb{R}^{2}$ bounded by the arcs $K\ OC,$ $K\ CF,$ $K\ OE,$ $K\ EF.$

######  Definition 4.

[11, p. 333] A set $D$ in $\mathbb{S}_{K}^{N},$ is convex if for any $X,Y\in
D$ there exists a geodesic arc $XY$ in $D$ such that $XY$ is the unique
minimizer in $\mathbb{S}_{K}^{N},$ connecting $X$ to $Y.$

######  Definition 5.

[11, p. 335] For any $X\ in\ \mathbb{S}_{K}^{N},$ the convexity radius is
given by: $ConvB(X,r)=\sup\\{\rho:B(X,r)\ is\ convex\ for\ all\ r<\rho,$ where
$B(X,r)$ is a disk with center at $X$ and radius $r.$

######  Lemma 6.

Convexity Radius for the spherical space $\mathbb{S}_{K}^{N},$ [33, p.
510],[19, Appendix,Lemma 1] The convexity radius $ConvB(X,r)$ for each
$X\in\mathbb{S}_{K}^{N},$ is $\frac{\pi}{4\sqrt{K}}.$

######  Theorem 7 (The weighted Fermat-Frechet problem in
$\mathbb{S}_{K}^{N}$).

The weighted Fermat-Frechet solution for a given $\frac{N(N+1)}{2}-$tuple
$a_{ij},$ such that $s\leq a_{ij}\leq\ell,$ for $(K\ell,Ks)\in
DW_{\mathbb{S}_{K}^{N}}(K\ell,Ks)$ and $\ell\leq\frac{\pi}{4\sqrt{K}}$
consists of a maximum number of $\frac{\frac{1}{2}N(N+1)!}{(N+1)!}$ weighted
Fermat trees, which belong to one of the following two cases:

(I) If for each index $k\in\\{1,2,3,\cdots,N\\},$

$\sum_{i=1,i<j}^{N}B_{i}B_{j}\frac{-\cos Ka_{ik}\cos Ka_{jk}+\cos
Ka_{ij}}{\sin Ka_{ik}\sin
Ka_{jk}}>\frac{B_{k}^{2}-\sum_{i=1,i<j}^{N}B_{i}^{2}}{2},$ (2.16)

we obtain the weighted floating Fermat-Torricelli tree
$\\{a_{01},a_{02},\cdots,a_{0N}\\}.$

(II) If there is an index $k\in{1,2,\cdots,N},$ such that:

$\sum_{i=1,i<j}^{N}B_{i}B_{j}\frac{-\cos Ka_{ik}\cos Ka_{jk}+\cos
Ka_{ij}}{\sin Ka_{ik}\sin
Ka_{jk}}\leq\frac{B_{k}^{2}-\sum_{i=1,i<j}^{N}B_{i}^{2}}{2},$ (2.17)

we obtain the weighted absorbing Fermat-Torricelli tree
$\\{a_{k1},a_{k2},\cdots,a_{kN}\\}.$

###### Proof.

The existence and uniqueness of the weighted Fermat-Torricelli point $A_{0}$
in $\mathbb{S}_{K}^{N}$ is given by compactness and the convexity of the
distance function $a_{0i}$ in [33, (1.2.4)], by selecting incongruent
$N-$simplexes $A_{1}A_{2}\cdots A_{N}$ in $B_{A_{i},\frac{\pi}{\sqrt{K}}}.$
Therefore, the maximum number of weighted Fermat trees for $s\leq
a_{ij}\leq\ell:$ $(Ks,K\ell)\in DW_{\mathbb{S}_{K}^{N}}(K\ell,Ks)$ is
$\frac{\frac{1}{2}N(N+1)!}{(N+1)!}.$

The gradient of the objective function
$f(a_{10},a_{20},\cdots,a_{N0})=\sum_{i=1}^{N}B_{i}a_{0i}$ gives a weighted
sum of outward pointing unit tangent vectors at $A_{0}$ $\nabla
f(a_{10},a_{20},\cdots,a_{N0})=\sum_{i=1}^{N}B_{i}\exp_{A_{0}}^{-1}\frac{\vec{X}(A_{0},A_{i})}{a_{0i}}.$
Taking the norm of the inner product $\nabla
f(a_{10},a_{20},\cdots,a_{N0})\dot{\exp}_{A_{0}}^{-1}\frac{\vec{X}(A_{0},A_{i})}{a_{0i}}$
yields an extension of weighted norm conditions of Kupitz-Martini for the
spherical case. By substituting the spherical law of cosines in in $\triangle
A_{i}AjA_{k}$ in the weighted norm inequalities of unit tangent vectors, we
get (2.16) and (2.17). ∎

In [7], Boroczsky proved an isoperimetric inequality of spherical simplexes in
$\mathbb{S}_{K}^{N}.$

######  Lemma 7.

Isoperimetric inequality of a spherical regular simplex in
$\mathbb{S}_{K}^{N},$ [7] A spherical $N$ simplex is of maximal volume if it
is regular.

######  Theorem 8 (The spherical Fermat-Torricelli Frechet solution).

The following equations depending on the variable spherical edge lengths
$a_{01},a_{02},\cdots,a_{0N}$ and the spherical edge lengths $\\{a_{ij}\\}$
provide a necessary condition for the location of the weighted floating
Fermat-Torricelli trees $\\{a_{01},a_{02},\cdots,a_{0N}\\},$ which belong to
the weighted Fermat-Frechet multitree solution:

$\displaystyle\sum_{i=2}^{N}\frac{B_{i}}{\sin Ka_{01}\sin Ka_{0i}}(-\cos
Ka_{01}\cos Ka_{0i}+\cos Ka_{1i})-$ $\displaystyle\sum_{i=1,{i\neq
j}}^{N}\frac{B_{i}}{\sin Ka_{0j}\sin Ka_{0i}}(-\cos Ka_{0j}\cos Ka_{0i}+\cos
Ka_{ji})=0,$ (2.18)

$\operatorname{Vol}(A_{1},A_{2},\cdots,A_{0}\cdots,A_{N}))<\operatorname{Vol}(C_{1},C_{2},\cdots,C_{N}))$
(2.19)

where $C_{1},C_{2},\cdots C_{N}$ is a regular spherical $N$-simplex with edge
length $c_{ij}=\frac{\sum_{i,j}a_{ij}}{\frac{(N-1)N}{2}}.$

###### Proof.

By differentiating the objective function $f(a_{01},a_{02},\cdots
a_{0N})=\sum_{i=1}^{N}B_{i}$ with respect to arc length and by using the first
variation formula of a variable arc length $a_{0i}$ with respect to arc length
$a_{01}$, and with respect to $a_{0j}$ and by subtracting the two derived
relations and then substituting $\cos\angle A_{i}A_{0}A_{j}$ by the spherical
law of cosines in $\triangle A_{i}A_{0}A_{j},$ we get (8). Taking into account
Lemma 7, we obtain :

$\operatorname{Vol}(A_{1},A_{2},\cdots,A_{0}\cdots,A_{N}))<\operatorname{Vol}(A_{1},A_{2},\cdots,A_{N}))<$
$\operatorname{Vol}(C_{1},C_{2},\cdots,\cdots,C_{N})),$

which gives (2.19).

∎

Given $\\{a_{01},a_{02},\cdots,a_{0N}\\}$ a weighted (floating) Fermat-
Torricelli tree with respect to the spherical simplex $A_{1}A_{2}\cdots A_{N}$
in $\mathbb{S}_{K}^{N}$ with unknown constant curvature $K,$ which belongs to
the spherical weighted Fermat-Frechet multitree, we may compute the curvature
$K$ and the radius $R$ of $\mathbb{S}_{K}^{N},$ by using the spherical Caley-
Menger determinant. The spherical Caley Menger determinant for
$\\{A_{0},A_{1},\cdots,A_{N}\\}$ is given by:

$\det(A_{0},A_{1},A_{2},\cdots,A_{N})=\\\ \begin{vmatrix}1&\cos Ka_{01}&\cos
Ka_{02}&\cdots&\cos Ka_{0N}\\\ \cos Ka_{10}&1&\cos Ka_{12}&\cdots&\cos
Ka_{1N}\\\ \cos Ka_{20}&\cos Ka_{21}&1&\cdots&\cos Ka_{2N}\\\ .&.&.&.&.&.\\\
\cos Ka_{N0}&\cos Ka_{N1}&\cos Ka_{N2}&\cdots&1\end{vmatrix}.$

By setting $\det(A_{0},A_{1},A_{2},\cdots,A_{N})=0,$ we derive proposition 3,
which gives an estimate of $K,$ (see in [45] for $N=3.$)

######  Proposition 3.

If a weighted Fermat-Frechet multitree $\\{a_{01},a_{02},\cdots,a_{0N}\\}$ is
given in $\mathbb{S}_{K}^{N}$ then the intersection of the solution set
$\det(A_{0},A_{1},A_{2},\cdots,A_{N})=0,$ for all incongruent spherical
simplexes with respect to $K$ gives the same sphere with radius
$\frac{1}{\sqrt{K}}.$

##  3\. A controlled Godel-Schoenberg’s isometric immersion of weighted Fermat
trees in $\mathbb{S}_{\rho_{1}}^{N-2}$ to a weighted simplex in
$\mathbb{S}_{\rho_{0}}^{N-1}$

In this section, we use Godel Schoenberg’s techniques, in order to construct
isometric immersions of weighted Fermat trees in $\mathbb{S}_{\rho_{1}}^{N-2}$
to a weighted simplex in $\mathbb{S}_{\rho_{0}}^{N-1}.$

In [23] (see also in [41, Footnote 5, p. 730]), Godel considered the edges of
a tetrahedron in $\mathbb{R}^{3}$ to be made of flexible strings and placed in
the interior of the tetrahedron a small sphere, which was gradually inflated.
After time $t,$ this sphere will become tightly packed within the six edges of
the tetrahedron. We are interested in the case where the three edges of the
tetrahedron correspond to a boundary spherical triangle and the other edges
are the three branches of the weighted Fermat (geodesic) tree, which meet at
an interior Fermat point.

Let $\triangle A_{1}A_{2}A_{3}$ be a spherical triangle in the open hemisphere
$\mathbb{S}_{K}^{2}$ of radius $R=\frac{1}{\sqrt{K}}\equiv\frac{1}{\kappa}.$
We denote by $A_{0}$ the weighted Fermat point inside $\triangle
A_{1}A_{2}A_{3}$ $\alpha_{ijk}\equiv\angle A_{i}A_{j}A_{k}$ and by
$\\{A_{0}A_{1},A_{0}A_{2},A_{0}A_{3}\\}$ the weighted Fermat tree, such that a
positive real number (weight) $B_{r}$ corresponds to each geodesic branch
$A_{0}A_{r},$ for $i,j,k=0,1,2,3,r=1,2,3.$

######  Theorem 9 (Godel’s isometric immersion of a weighted Fermat tree with
respect to a boundary spherical triangle to a weighted tetrahedron in
$\mathbb{R}^{3}$).

If we select a triad of weights $\\{B_{1},B_{2},B_{3}\\},$ which satisfy

$\displaystyle 1-(\frac{\sin(\kappa
a_{13})}{\sin(\arccos(\frac{B_{2}^{2}-B_{1}^{2}-B_{3}^{2}}{2B_{1}B_{3}}))}\sin(\alpha_{213}-\arccos(\frac{\cos(\kappa
a_{02})-\cos(\kappa a_{01})\cos(\kappa a_{12})}{\sin(\kappa a_{01})\sin(\kappa
a_{12})}))))^{2}={}$ (3.1) $\displaystyle{}(\cos(\kappa a_{01})\cos(\kappa
a_{13})+\sin(\kappa a_{13})\cos(\alpha_{213})\frac{\cos(\kappa
a_{02})-\cos(\kappa a_{01})\cos(\kappa a_{12})}{\sin(\kappa a_{12})}+{}$
$\displaystyle{}+\sin(\kappa a_{01})\sin(\kappa
a_{13})\sin(\alpha_{213})\cdot{}$
$\displaystyle{}\cdot\sqrt{1-(\frac{\cos(\kappa a_{02})-\cos(\kappa
a_{01})\cos(\kappa a_{12})}{\sin(\kappa a_{01})\sin(\kappa
a_{12})})^{2}}\quad\quad\quad)^{2},$

$(\cos(\kappa a_{12})-\cos(\kappa a_{01})\cos(\kappa a_{02}))^{2}=(\sin(\kappa
a_{01})\sin(\kappa
a_{02})\frac{B_{3}^{2}-B_{1}^{2}-B_{2}^{2}}{2B_{1}B_{2}})^{2}$ (3.2)

such that $\\{a_{01},a_{02},a_{03},a_{12},a_{13},a_{23}\\}\in
DW_{\mathbb{R}^{2}}(\ell,s)$ then the weighted Fermat tree
$\\{a_{01},a_{02},a_{03}\\}$ and the edges of the boundary spherical triangle
are isometrically immersed to a weighted tetrahedron in $\mathbb{R}^{3}$ with
corresponding edges $\\{a_{01},a_{02},a_{03},a_{12},a_{13},a_{23}\\}$ and
corresponding weights $\\{B_{1},B_{2},B_{3},1,1,1\\}.$

###### Proof.

We construct an isometric immersion of a Fermat tree with respect to
$\triangle A_{1}A_{2}A_{2}$ following Godel’s observation of a tetrahedron
with six flexible edges tightly pack to a sphere of given radius
$R=\frac{1}{\sqrt{K}}=\frac{1}{\kappa}$ and taking into account two equations
derived by [50, Theorem 2.4, (2.23),(2.24)], which determine the location of
the weighted Fermat-tree. By restricting the edge lengths
$\\{a_{01},a_{02},a_{03},a_{12},a_{13},a_{23}\\}\in
DW_{\mathbb{R}^{2}}(\ell,s),$ we obtain a weighted tetrahedron
$A_{0}A_{1}A_{2}A_{3}$ in $\mathbb{R}^{3}.$ ∎

######  Remark 1.

We note that the two equations (3.1) and (3.2) , which give the location of
the weighted Fermat point $\triangle A_{1}A_{2}A_{3}$ on the K-plane (two
dimensional sphere $S_{K}^{2}$ and two dimensional hyperbolic plane
$H_{K}^{2}$) can be merged to a rational algebraic equation, which depends
only on the variable $z\equiv\sin\angle A_{0}A_{1}A_{3}$ (see in [51, Theorem
2.4, (2.8)]).

In [41, Theorem 3,p. 728], Schoenberg obtained an isometric immersion of an
$(N-1)$ spherical simplex in $\mathbb{S}_{\rho_{0}}^{N-1}$ to
$\mathbb{S}_{\rho_{1}}^{N-2}$ by proving the existence of a radius
$\rho_{1}\leq\rho_{0}.$ Thus, by applying Schoenberg’s isometric immersion, we
derive the following reformulation of Theorem 9 for $N=4.$

######  Theorem 10 (Schoenberg’s isometric immersion of a weighted Fermat tree
with respect to a boundary spherical triangle in $\mathbb{S}_{\rho_{1}}^{2}$
to a weighted tetrahedron in $\mathbb{S}_{\rho_{0}}^{3}$).

If we select a triad of weights $\\{B_{1},B_{2},B_{3}\\},$ which satisfy

$\displaystyle
1-(\frac{\sin(\kappa_{1}a_{13})}{\sin(\arccos(\frac{B_{2}^{2}-B_{1}^{2}-B_{3}^{2}}{2B_{1}B_{3}}))}\sin(\alpha_{213}-\arccos(\frac{\cos(\kappa_{1}a_{02})-\cos(\kappa_{1}a_{01})\cos(\kappa_{1}a_{12})}{\sin(\kappa_{1}a_{01})\sin(\kappa_{1}a_{12})}))))^{2}={}$
(3.3)
$\displaystyle{}(\cos(\kappa_{1}a_{01})\cos(\kappa_{1}a_{13})+\sin(\kappa_{1}a_{13})\cos(\alpha_{213})\frac{\cos(\kappa_{1}a_{02})-\cos(\kappa_{1}a_{01})\cos(\kappa_{1}a_{12})}{\sin(\kappa_{1}a_{12})}+{}$
$\displaystyle{}+\sin(\kappa_{1}a_{01})\sin(\kappa_{1}a_{13})\sin(\alpha_{213})\cdot{}$
$\displaystyle{}\cdot\sqrt{1-(\frac{\cos(\kappa_{1}a_{02})-\cos(\kappa_{1}a_{01})\cos(\kappa_{1}a_{12})}{\sin(\kappa_{1}a_{01})\sin(\kappa_{1}a_{12})})^{2}}\quad\quad\quad)^{2},$

$(\cos(\kappa_{1}a_{12})-\cos(\kappa_{1}a_{01})\cos(\kappa_{1}a_{02}))^{2}=(\sin(\kappa_{1}a_{01})\sin(\kappa_{1}a_{02})\frac{B_{3}^{2}-B_{1}^{2}-B_{2}^{2}}{2B_{1}B_{2}})^{2}$
(3.4)

then there exists a $\rho_{0}>\rho_{1},$ such that the weighted Fermat tree
$\\{a_{01},a_{02},a_{03}\\}$ and the edges of the boundary spherical triangle
in $\mathbb{S}_{\rho_{1}}^{2}$ are isometrically immersed to a weighted
spherical tetrahedron in $\mathbb{S}_{\rho_{0}}^{3}$ with corresponding edges
$\\{a_{01},a_{02},a_{03},a_{12},a_{13},a_{23}\\}$ and corresponding weights
$\\{B_{1},B_{2},B_{3},1,1,1\\},$

for $\\{a_{01},a_{02},a_{03},a_{12},a_{13},a_{23}\\}\in
DW_{\mathbb{S}_{\rho_{0}}^{3}}(\ell,s).$

We note that we may use the weighted Fermat-Torricelli tree for an equilateral
triangle having equal edge lengths $\frac{\pi}{2}$ in $\mathbb{S}_{1}^{2},$
which was given explicitly in [52], in order to construct a weighted isometric
immersion in $\mathbb{S}_{\rho_{0}}^{3}$ and in $\mathbb{R}^{3}.$ These
isometric immersions need to be controlled by the conditions for the edge
lengths described by the Dekster-Wilker spherical and Euclidean domain.

Let $A_{1}A_{2}\cdots A_{N-1}$ be an $(N-2)$-simplex in
$\mathbb{S}_{\rho_{1}}^{N-2},$ $A_{0}$ be the weighted Fermat point of the
$(N-2)$-simplex having degree $N-1$ and
$\\{A_{1}A_{0},A_{2}A_{0},\cdots,A_{N-1}A_{0}\\}$ the corresponding weighted
Fermat tree. We denote by $B_{i}$ the weight, which corresponds to each
geodesic branch $A_{i}A_{0}$ and by $a_{jm}$ the length of the geodesic arc
$A_{j}A_{m},$ for $i=0,1,2,\cdots N-1$ and $j,m=0,1,2,\cdots,N-1.$

We denote by
$\\{A_{1},A_{2},\cdots,A_{0}\cdots,A_{N-1}\\}\subset\\{A_{1},A_{2},\cdots,A_{N-1}\\}$
an $(N-2)$-simplex, which is derived by replacing the vertex
$A_{j}\in\\{A_{1},A_{2},\cdots,A_{N-1}\\}$ with $A_{0}.$

######  Theorem 11 (Godel-Schoenberg’s isometric immersion of weighted Fermat
trees in $\mathbb{S}_{\rho_{1}}^{N-2}$ to $\mathbb{S}_{\rho_{0}}^{N-1}$).

If we select an $(N-1)$-tuple of weights $\\{B_{1},\cdots,B_{N-1}\\},$ which
satisfy

$\displaystyle\sum_{i=2}^{N-1}\frac{B_{i}}{\sin Ka_{01}\sin Ka_{0i}}(-\cos
Ka_{01}\cos Ka_{0i}+\cos Ka_{1i})-$ $\displaystyle\sum_{i=1,{i\neq
j}}^{N-1}\frac{B_{i}}{\sin Ka_{0j}\sin Ka_{0i}}(-\cos Ka_{0j}\cos Ka_{0i}+\cos
Ka_{ji})=0,$ (3.5)

$\operatorname{Vol}(A_{1},A_{2},\cdots,A_{0}\cdots,A_{N-1}))<\operatorname{Vol}(C_{1},C_{2},\cdots,C_{N-1}))$
(3.6)

where $C_{1},C_{2},\cdots C_{N-1}$ is a regular spherical $(N-2)$-simplex with
edge length $c_{ij}=\frac{\sum_{i,j}a_{ij}}{\frac{(N-2)(N-1)}{2}}.$

then there exists a $\rho_{0}>\rho_{1},$ such that the weighted Fermat tree
$\\{a_{01},\cdots,a_{0N-1}\\}$ and the edges of the boundary $(N-1)$-simplex
$\\{a_{ij}\\}$ in $\mathbb{S}_{\rho_{1}}^{N-2}$ are isometrically immersed to
a weighted spherical $(N-1)$-simplex in $\mathbb{S}_{\rho_{0}}^{N-1}$ with
corresponding edges $\\{a_{01},\cdots,a_{0N-1},\\{a_{ij}\\}\\}$ and
corresponding weights $\\{B_{1},B_{2},\cdots,B_{N-1},1,\cdots,1\\},$

for the $\frac{(N-1)N}{2}$tuple of edge lengths
$\\{a_{01},\cdots,a_{0N-1},\\{a_{ij}\\}\\}\in
DW_{\mathbb{S}_{\rho_{0}}^{N-1}}(\ell,s).$

###### Proof.

By applying Schoenberg isometric immersion and by taking into account the
conditions of Theorem 8, we can construct an isometric immersion od weighted
Fermat trees in $\mathbb{S}_{\rho_{1}}^{N-2}$ to
$\mathbb{S}_{\rho_{0}}^{N-1},$ the $\frac{(N-1)N}{2}$tuple of edge lengths
$\\{a_{01},\cdots,a_{0N-2},\\{a_{ij}\\}\\}\in
DW_{\mathbb{S}_{\rho_{0}}^{N-1}}(\ell,s).$ ∎

By substituting $\rho=\infty$ in Schoenberg’s isometric immersion, we get
Godel-Schoenberg’s isometric immersion ([41, Theorem 3’,p. 730] ), we derive
Godel’s isometric immersion of weighted Fermat trees for a boundary
$(N-1)$-simplex in $\mathbb{S}_{\rho}^{N-1}$ to a weighted $N$-simplex
$\mathbb{R}^{N}.$

We denote by
$\\{A_{1},A_{2},\cdots,A_{0}\cdots,A_{N}\\}\subset\\{A_{1},A_{2},\cdots,A_{N}\\}$
an $(N-1)$-simplex, which is derived by replacing the vertex
$A_{j}\in\\{A_{1},A_{2},\cdots,A_{N}\\}$ with $A_{0}.$

######  Theorem 12 (Godel’s isometric immersion of weighted Fermat trees in
$\mathbb{S}_{\rho}^{N-1}$ to $\mathbb{R}^{N}$).

If we select an $N$-tuple of weights $\\{B_{1},\cdots,B_{N-1},B_{N}\\},$ which
satisfy

$\displaystyle\sum_{i=2}^{N}\frac{B_{i}}{\sin Ka_{01}\sin Ka_{0i}}(-\cos
Ka_{01}\cos Ka_{0i}+\cos Ka_{1i})-$ $\displaystyle\sum_{i=1,{i\neq
j}}^{N}\frac{B_{i}}{\sin Ka_{0j}\sin Ka_{0i}}(-\cos Ka_{0j}\cos Ka_{0i}+\cos
Ka_{ji})=0,$ (3.7)

$\operatorname{Vol}(A_{1},A_{2},\cdots,A_{0}\cdots,A_{N}))<\operatorname{Vol}(C_{1},C_{2},\cdots,C_{N-1},C_{N})$
(3.8)

where $\\{C_{1},C_{2},\cdots C_{N-1},C_{N}\\}$ is a regular spherical
$(N-1)$simplex with edge length
$c_{ij}=\frac{\sum_{i,j}a_{ij}}{\frac{(N-1)N}{2}}.$

then there exists an isometric immersion of the weighted Fermat tree
$\\{a_{01},\cdots,a_{0N}\\}$ and the edges of the boundary $N-1$ simplex
$\\{a_{ij}\\}$ in $\mathbb{S}_{\rho_{1}}^{N-1}$ to a weighted $N$-simplex in
$\mathbb{R}^{N}$ with corresponding edges
$\\{a_{01},\cdots,a_{0N},\\{a_{ij}\\}\\}$ and corresponding weights
$\\{B_{1},B_{2},\cdots,B_{N},1,\cdots,1\\},$

for the $\frac{(N(N+1}{2}$tuple of edge lengths
$\\{a_{01},\cdots,a_{0N},\\{a_{ij}\\}\\}\in DW_{\mathbb{R}^{N}}(\ell,s).$

We consider a $\frac{(N-1)N}{2}-$tuple of positive real numbers $a_{ij},$
determining the edge lengths of $(N-1)$ incongruent spherical simplexes, such
that: $a_{ij}$ belong to the spherical domain of Dekster-Wilker
$DW_{\mathbb{S}_{\rho_{0}}^{N-1}}(\ell,s)$ and $s\leq a_{ij}\leq\ell.$ Thus,
all incongruent pairwise spherical $(N-1)$-simplexes may yield up to
$\frac{\frac{1}{2}(N-1)N!}{N!}$ weighted Fermat-Torricelli trees for a given
$N$-tuple of weights $\\{B_{1},B_{2},\cdots,B_{N}.$ The union of these
weighted Fermat-Torricelli trees determine the weighted Fermat-Frechet
multitree (solution).

######  Definition 6 (Stable isometric immersion of a variable weighted
Fermat-Frechet multitree).

We call a stable isometric immersion of a variable weighted Fermat-Frechet
multitree for a $\frac{(N-1)N}{2}-$tuple of positive real numbers $a_{ij},$
determining $(N-1)-$ incongruent spherical simplexes in
$\mathbb{S}_{\rho}^{N-1}$ to $\mathbb{R}^{N}$ the solution set of the variable
weights $\\{B_{1},B_{2},\cdots,B_{N}\\},$ such that each $N-1$ spherical
simplex with the corresponding variable weighted Fermat tree in
$\mathbb{S}_{\rho}^{N-1}$ is isometrically immersed to an $N-$simplex in
$\mathbb{R}^{N}.$

######  Definition 7 (Godel-Frechet multisimplex).

We call Godel-Frechet multisimplex a union of $N$-simplexes in
$\mathbb{R}^{N},$ which is derived by a stable isometric immersion of a
variable weighted Fermat-Frechet multitree in $\mathbb{S}_{\rho}^{N-1}.$

We denote by $B_{i}(t)$ a variable weight, which correspond to the geodesic
branch $A_{0}A_{i}$ of each weighted Fermat tree
$\\{A_{1}A_{0},c\dots,A_{N}A_{0}\\},$ which forms a weighted Fermat-Frechet
multitree, for $t>0,$ $i=1,2,\cdots,N.$

######  Theorem 13 (Godel’s isometric immersions of a weighted Fermat-Frechet
multitree in $\mathbb{S}_{\rho}^{N-1}$ to $\mathbb{R}^{N}$ ).

The following conditions for $B_{i}(t)$ provide a stable isometric immersion
of a weighted Fermat-Frechet multisimplex in $\mathbb{S}_{\rho}^{N-1}$ to a
Godel-Frechet simplex in $\mathbb{R}^{N}:$

$\displaystyle\sum_{i=2}^{N}\frac{B_{i}(t)}{\sin Ka_{01}\sin Ka_{0i}}(-\cos
Ka_{01}\cos Ka_{0i}+\cos Ka_{1i})-$ $\displaystyle\sum_{i=1,{i\neq
j}}^{N}\frac{B_{i}(t)}{\sin Ka_{0j}\sin Ka_{0i}}(-\cos Ka_{0j}\cos
Ka_{0i}+\cos Ka_{ji})=0,$ (3.9)

$\sum_{i=1}^{N}B_{i}(t)=1,$ (3.10)

$\operatorname{Vol}(A_{1},A_{2},\cdots,A_{0}\cdots,A_{N}))<\operatorname{Vol}(C_{1},C_{2},\cdots,C_{N}))$
(3.11)

where $C_{1},C_{2},\cdots C_{N}$ is a regular spherical $(N-1)$simplex with
edge length $c_{ij}=\frac{\sum_{i,j}a_{ij}}{\frac{(N-1)N}{2}}.$

for the $\frac{(N(N+1}{2}$tuple of edge lengths
$\\{a_{01},\cdots,a_{0N},\\{a_{ij}\\}\\}\in DW_{\mathbb{R}^{N}}(\ell,s).$

###### Proof.

By substituting $B_{i}(t)\to B_{i}$ in (12) taking into account the
isoperimetric condition for the variable weights (3.10) and the volume
inequality (3.8) taken from Theorem 12, we derive (13), which yields the
desired stable isometric immersion of the variable weighted Fermat-Frechet
spherical multitree to the variable weighted Godel-Frechet multisimplex in
$\mathbb{R}^{N}.$ ∎

We focus on small perturbations $\epsilon_{ij}$ of a $\frac{(N-1)N}{2}-$tuple
of positive real numbers $a_{ij},$ determining the edge lengths of $(N-1)$
incongruent spherical simplexes, such that: $a_{ij},a_{ij}+\epsilon_{ij}\in
DW_{\mathbb{S}_{\rho_{0}}^{N-1}}(\ell,s)$ and $s\leq a_{ij}\leq\ell,$ under
the condition:

$\sum_{i,j=1,i<j}^{N}a_{ij}=\sum_{i,j=1,i<j}^{N}a_{ij}+\epsilon_{ij},$

for $\|\epsilon_{ij}\|<<1$ and $\sum_{i,j=1,i<j}^{N}\epsilon_{ij}=0.$

By substituting $a_{ij}+\epsilon_{ij}\to a_{ij}$ in Theorem 13, we derive the
conditions to create a family of stable isometric immersions of spherical
variable weighted Fermat-Frechet multitrees to Godel-Frechet multisimplexes
depending on the isoperimetric perturbations $\epsilon_{ij}$ of the initial
boundary simplex in $\mathbb{S}_{\rho_{0}}^{N-1}.$

######  Theorem 14 (Stable isometric immersions of weighted Fermat-Frechet
multitrees for isoperimetric deformations of the boundary $N-$simplex in
$\mathbb{S}_{\rho_{0}}^{N-1}$ to $\mathbb{R}^{N}$ ).

The following conditions for $B_{i}(t)$ provide a family of stable isometric
immersions of variable weighted Fermat-Frechet multisimplexes in
$\mathbb{S}_{\rho}^{N-1}$ to $\epsilon_{ij}$-Godel-Frechet simplexes in
$\mathbb{R}^{N}:$

$\displaystyle\sum_{i=2}^{N}\frac{B_{i}(t)}{\sin Ka_{01}\sin Ka_{0i}}(-\cos
Ka_{01}\cos Ka_{0i}+\cos K(a_{1i}+\epsilon_{1i})-$
$\displaystyle\sum_{i=1,{i\neq j}}^{N}\frac{B_{i}(t)}{\sin Ka_{0j}\sin
Ka_{0i}}(-\cos Ka_{0j}\cos Ka_{0i}+\cos K(a_{ji}+\epsilon_{ji})=0,$ (3.12)

$\sum_{i=1}^{N}B_{i}(t)=1,$ (3.13)

$\operatorname{Vol}(A_{1},A_{2},\cdots,A_{0}\cdots,A_{N})(\\{\epsilon_{ij}\\})<\operatorname{Vol}(C_{1},C_{2},\cdots,C_{N}))(\\{\epsilon_{ij}\\})$
(3.14)

where $C_{1},C_{2},\cdots C_{N}$ is a regular spherical $(N-1)$simplex with
edge length $c_{ij}=\frac{\sum_{i,j}a_{ij}+\epsilon_{ij}}{\frac{(N-1)N}{2}}.$

for the $\frac{(N(N+1}{2}$tuple of edge lengths
$\\{a_{01},\cdots,a_{0N},\\{a_{ij}\\}\\}\in DW_{\mathbb{R}^{N}}(\ell,s).$

##  4\. Gromov isometries for the Fermat-Steiner-Frechet solution in
$\mathbb{H}^{N}$

In this section, we introduce an embedding (inclusion map) of a Fermat-
Steiner-Frechet multitree solution for a given $\frac{N(N+1}{2}$-tuple of edge
lengths determining incongruent boundary $N$-simplexes to an associated family
of Gromov isometries up to an additive constant for $N$-simplexes and ideal
$N$-simplexes in $\mathbb{H}_{K}^{N}.$

We proceed by giving the definitions of an ideal $k$-simplex
$\triangle_{i}^{k}\equiv A_{1}^{i}A_{2}^{i}\cdots A_{k+1}^{i}$ in
$\mathbb{H}_{N}^{k}$ with constant curvature $K=-\epsilon^{2}<0,$ a class of
geodesic trees enriched by some properties given by Gromov (intermediate
geodesic Fermat-Steiner trees) and Gromov’s (+) isometry (up to an additive
constant).

The hyperbolic $N$ space $\mathbb{H}_{N}^{k}$ may be represented as the
Poincare disc model or the half space model or the projective (Klein) model.

If we consider that $\mathbb{H}_{N}$ is represented by the Poincare disc model

$\mathbb{H}^{N}\approx D^{N}=\\{x\in\mathbb{R}^{N}|||x||<1\\}$

with the Riemannian metric

$ds^{2}=\frac{4}{1-r^{2}}\sum_{i=1}^{N}(dx_{i})^{2},\
r^{2}\equiv\sum_{i=1}^{N}x_{i}^{2}$

(see in [26]).

######  Definition 8.

[26, p. 1] The hyperbolic boundary $\partial\mathbb{H}^{N}$ is the ”sphere at
infinity”

$\partial\mathbb{H}^{N}=\\{x\in\mathbb{R}^{N}|||x||=1\\}=\mathbb{S}^{N-1}.$

######  Definition 9.

An ideal $k$-simplex $\triangle_{i}^{k},$ [24, p. 233] An ideal $N$-simplex in
$\mathbb{H}_{-\epsilon^{2}}^{N}$ is the convex hull of $N+1$ distinct points
(vertices)
$\\{A_{1}^{id},A_{2}^{id},\cdots,A_{N}^{id},A_{N+1}^{id}\\}$ in the hyperbolic
boundary $\partial\mathbb{H}_{-\epsilon^{2}}^{N}$ (”sphere at infinity”).

In [46], Thurston showed that all ideal triangles in $\mathbb{H}_{-1}^{2}$ are
isometric. In [37] Milnor derived that ideal hyperbolic tetrahedra are not
mutually isometric, In [24][pp. 233-234]Gromov assigned properties in a
geodesic tree $S\subset\Delta^{N}\subset\mathbb{H}^{N}$ and used in a
canonical way an inclusion map $S\to\Delta^{N},$ to confirm that ideal
$N-$simplexes are not mutually isometric for $N\geq 3.$

We extend the definitions of Steiner tree topologies given in [22] and
intermediate Fermat-Steiner tree topologies, which have been introduced in
[54] in $\mathbb{R}^{3},$ for geodesic trees in
$\mathbb{H}_{-\epsilon^{2}}^{N}.$

######  Definition 10 (Geodesic tree topology).

A geodesic tree topology is a connection matrix specifying which pairs of
points from the list
$\\{A_{1},A_{2},\cdots A_{N},X_{1},X_{2},\cdots
X_{N-2}\\}\in\mathbb{H}_{-\epsilon^{2}}^{N}$ have a connecting geodesic
segment (edge).

######  Definition 11 (Degree of a vertex).

The degree of a vertex corresponds to the number of connections of the vertex
with geodesic segments.

######  Definition 12 (Degree of an intermediate Fermat-Steiner point).

The degree of a (weighted) intermediate Fermat-Steiner point (vertex) $X$ with
respect to a boundary simplex $A_{1},A_{2},\cdots
A_{N}\in\mathbb{H}_{-\epsilon^{2}}^{N}$ is greater or equal than $3$ and less
or equal than $N.$

######  Definition 13 (Fermat (geodesic) tree topology).

A Fermat tree topology is a tree topology with a connection matrix
$A_{1},A_{2},\cdots A_{N},X_{1},$ such that the boundary vertices
$A_{1},A_{2},\cdots A_{N}$ have degree $1$ and the Fermat-point $X_{1}$ has
degree $N$.

######  Definition 14 (Fermat-Steiner (geodesic)) tree topology).

A (full) Fermat-Steiner tree topology is a tree topology with a connection
matrix $A_{1},A_{2},\cdots A_{N},X_{1},X_{2},\cdots X_{N-1},$ such that the
boundary vertices of the simplex $A_{1},A_{2},\cdots A_{N}$ have degree $1$
and the Fermat-Steiner points $X_{1},X_{2},\cdots X_{N-1}$ have degree $3.$

######  Definition 15 (Intermediate Fermat-Steiner (geodesic) tree topology in
$\mathbb{H}_{k}^{N}$).

An intermediate Fermat-Steiner tree topology is a tree topology, which has
$m\leq N-2$ $X_{1},X_{2},\cdots X_{m}$ vertices inside the simplex
$A_{1},A_{2},\cdots A_{N}\in\mathbb{H}_{-\epsilon^{2}}^{N},$ such that the
boundary vertices of the simplex $A_{1},A_{2},\cdots A_{N}$ have degree $1$
and the intermediate Fermat-Steiner points $X_{1},X_{2},\cdots X_{m}$ have
degree less than $N.$

The intermediate Fermat-Steiner tree $S$ is a union of some geodesic segments
between the intermediate Fermat-Steiner vertices $X_{j}$ and some geodesic
segments connecting each $A_{i}$ with some $X_{j}.$ In [24], Gromov assigned
in a canocical way an intermediate Fermat-Steiner geodesic tree for an ideal
simplex $A_{1},A_{2},\cdots A_{N}$ some geometric properties [24,
(a),(b),(c),(d),(e), p. 233].

######  Example 1.

An intermediate Fermat-Steiner tree $S$ of a $4$-simplex
$A_{1},A_{2},A_{3},A_{4},A_{5}$ associated with an intermediate Fermat-Steiner
tree topology is a collection of the geodesic segments
$\\{A_{1}X_{1},A_{4}X_{1},X_{1}X_{2},$
$A_{5}X_{2},A_{2}X_{2},A_{3}X_{2}\\}.$ The intermediate Fermat Steiner points
$X_{1},$ $X_{2}$ have degrees $3$ and $4,$ respectively and the boundary
vertices $\\{A_{1},A_{2},A_{3},$
$A_{4},A_{5}\\}$ have degree $1.$

######  Definition 16 (Intermediate Fermat-Steiner Frechet multitree).

An intermediate Fermat-Steiner Frechet multitree is a collection of
intermediate Fermat-Steiner trees, which correspond to incongruent
$N$-simplexes derived by a $\frac{N(N+1}{2}-$tuple of positive real numbers
determining edge lengths.

######  Remark 2.

An intermediate Fermat-Steiner Frechet multitree $MS$ coincides with a Fermat-
Frechet multitree, by setting only one vertex $X_{1}$ inside each derived
simplex (Frechet multisimplex $F\Delta^{N}$ ) having degree $N.$

######  Definition 17.

A Fermat-Frechet multispanning tree is a Fermat-Frechet multitree with zero
interior vertices ($m=0$) at each simplex, which belongs to the Frechet
multisimplex.

In [24, 6.2, pp. 157-158], Gromov constructed a geodesic tree ($\log_{2}N$
spanning geodesic tree) $\tau]subsetX,$ for a finite subset
$\\{V=V_{1},V_{2},\cdots,V_{N}\\}$ of a geodesic $\delta$ hyperbolic space
$\mathbb{X}$ with following properties:

(1) $\tau$ is a union of at most $N-1$ geodesic segments in $\mathbb{X}$ ,

(2) the set of extremal points of $\tau$ equals $V,$

(3) Every two points $V_{i},$ $V_{j}$ in $V$ can be joined by a broken
geodesic g having $k\leq 1+2log_{2}N$ segments, such that:

$lengthg\leq|V_{i}-V_{j}|+C\delta(log_{2}(N))^{2},$

for $C\leq 100.$

In [8], [9], Bowditch obtained another tree-like structure of a $\delta$
hyperbolic space $\mathbb{X}$ in the sense of Gromov, by determining an
isometry of a spanning tree up to an additive constant, which depend on $N$
and a constant $\delta,$ which lead to an isometric embedding of ”logarithmic
($\log N$) spanning trees”.

######  Lemma 8.

Isometric embedding of a spanning tree up to an additive constant, [9,
Proposition 6.7,pp. 49-51] There is a function $h:\mathbb{N}\to[0,\infty)$
such that if $F\subset\mathbb{X}$ with $|F|=\mathbb{N},$ then there is a
(spanning) tree $\tau,$ such that for all $X,Y\in F,$ $d_{\tau}(X,Y)\leq
XY+\delta h(N),$ ($d_{\tau}$ is distance measured in $\tau$ and all the edges
of $\tau$ are geodesic segments).

######  Lemma 9.

Isometric embedding of a spanning tree up to an additive logarithmic constant,
[8, Proposition 7.3.1, Theorem 7.6.1] There is a function
$f:\mathbb{N}\to\mathbb{R},$ such that the following holds. Suppose
($\mathbb{X},d)$ is a $\delta$ hyperbolic geodesic space and the vertex set
$V\subset\mathbb{X}$ is a set of $(N+1)$points. Then there is an immersed
spanning tree $\tau$ for $V$ in $\mathbb{X},$ such that for any $X,Y\in V,$ we
have

$\rho_{\tau}(X,Y))\leq d(X,Y)+hF(N),$

where $F(N)=O(logN).$

We proceed by obtaining an embedding of Fermat-Steiner Frechet multitree to
$\Delta^{N}$, taking into consideration Gromov’s observation that the regular
ideal simplex having the maximal volume corresponds to a tree $S$ having a
maximal length metric among the Fermat-Steiner trees, which consists of the
union of $N+1$ rays joining the ideal points $A_{i}^{\circ}$ with only a
single point $X_{1}\in\mathbb{H}^{N}$ ([24, Property (d),p. 233]).

We consider an example of a tree $S$ for a boundary tetrahedron in
$\mathbb{R}^{3}.$ We show that we can perturb the maximal length (metric) of
the unweighted Fermat- Steiner trees, by assigning properly some weights at
each vertex of the tetrahedron, which yields a maximal weighted length metric
of the Fermat tree greater than the maximal unweighted length metric of the
weighted Fermat tree having the same geometric structure (tree topology).

######  Example 2.

Let $A_{1}A_{2}A_{3}A_{4}$ be a tetrahedron and $X_{1}$ be the corresponding
Fermat point in $\mathbb{R}^{3}.$ One can rotate by a suitable angle (twist
angle), such that $A_{1}A_{2}A_{3}A_{4}$ and $X_{1}$ lie on the same plane.
Thus, $X_{1}$ is the intersection of the diagonals $A_{1}A_{3}$ and
$A_{2}A_{4}.$ One can assign the following weights $B_{i}$ (positive real
numbers) at each vertex $A_{i}$ (Fig. 1). The length of the unweighted
($B_{i}=1$) Fermat tree is given by

$lengthg=A_{1}A_{3}+A_{2}A_{4}.$

We can perturb the length metric of the Fermat tree, by assigning the
following weights at each $A_{i},$ such that the weighted Fermat point $X_{1}$
remains the same:

$B_{1}=B_{3}=1+\epsilon,\ B_{2}=B_{4}=1-\epsilon.$
$lengthg_{p}=(1+\epsilon)A_{1}A_{3}+(1-\epsilon)A_{2}A_{4}.$

Hence, we obtain the inequality for $\epsilon>0$ and $a_{13}>a_{24}$ or
$\epsilon<0$ and $a_{13}<a_{24}:$

$lengthg_{p}-lengthg=\epsilon(a_{13}-a_{24})>0.$

Figure 1.  Perturbing the Fermat length metric for $A_{1}A_{2}A_{3}A_{4}$ with
weights $B_{1}=B_{3}=1+\epsilon,\ B_{2}=B_{4}=1-\epsilon$

######  Theorem 15.

The embedding of an intermediate Fermat-Steiner Frechet multitree $MS\to
F\Delta^{N}$ is a collection of isometries up to an additive constant, where
the implied constants only depends on $N$ and $\epsilon.$

###### Proof.

A. A Proof using weights

We can assign weights $B\\{_{1},B_{2},\cdots B_{N+1}\\}$ to each vertex of the
simplex $A_{1}A_{2}\cdots A_{N+1}\in\mathbb{H}_{-\epsilon^{2}}^{N},$ to
perturb the corresponding maximal length metric of the Fermat-tree, which
gives an isometry up to an additive $\log_{2}N$ or $\log N$ constant due to
Gromov, Bowditch for a given $\frac{N(N+1}{2}$tuple of edge lengths
determining a simplex by using the solution domain of Dekster-Wilker
DW($\mathbb{H}_{\epsilon^{2}}^{N}$). Thus, the collection of the isometries up
to an additive constant depend on $\epsilon$ and $N$ and gives a
”multi-”inclusion map of intermediate Fermat-Steiner-Frechet multitrees $MS$
to a Frechet multisimplex $F\Delta^{N}.$

B. Proof without using weights

We use the following result for the length metric of minimal Steiner trees and
associated minimal spanning trees, which has been proved by Ivanov Tuzhilin
and Cieslik for manifolds ([31], [29]). The length of the Fermat-Steiner tree
with intermediate vertices is less than length of the spanning tree with zero
intermediate vertices. Therefore, by applying Gromov-Bowditch constructions,
we get

$length(Fermat\ Steiner\ tree)<length{Spanning\
tree}\leq|V_{i}-V_{j}|+C\delta(log_{2}(N))^{2},$

for $C\leq 100.$

∎

We define a dynamic intermediate Fermat-Steiner Frechet multitree for some
perturbed hyperbolic Frechet multsimplexes and we consider emmbeddings of
dynamic intermediate Fermat-Steiner Frechet multitrees to almost regular
Frechet multisimplex in terms of Dekster’s lower bound estimate for the
generalized Santalo’s thickness of simplexes in
$\mathbb{H}_{-\epsilon^{2}}^{N}$ ([16, Theorem 2 (2.3),pp.51 52]).

######  Definition 18 (A dynamic intermediate Fermat-Steiner Frechet
multitree).

A dynamic intermediate Fermat-Steiner-Frechet multitree is a union of Fermat-
Steiner-Frechet multitrees, which are derived by a given
$\frac{N(N+1)}{2}-$tuple of positive real numbers determined by the edge
lengths $\ell_{i}$ of a Frechet multisimplex in $\mathbb{H}_{k}^{N}$ perturbed
by a real number $\epsilon_{i}:$ $\ell_{i}^{\prime}=\ell_{i}+\epsilon_{i}:$

$\sum_{i=1}^{\frac{N(N+1)}{2}}\ell_{i}^{\prime}=\sum_{i=1}^{\frac{N(N+1)}{2}}\ell_{i}^{\prime},$

where $\sum_{i=1}^{\frac{N(N+1)}{2}}\epsilon_{i}=0$ and $\ell_{i}^{\prime}$
are the edge lengths of a Frechet simplex obtained by the Dekster Wilker
solution domain $DW(\mathbb{H}_{k}^{N}).$

We note that Dekster ([16, p 52, (1.14), Example,pp. 53-54]) extended Gromov’s
thinness of geodesic triangles in $\delta=\frac{1}{2\epsilon}\log 3$
hyperbolic spaces ([24, 6.3, Lemma 6.3.A],[9, Example 4,p. 54]), by obtaining
a generalization of Santalo’s thickness from $\mathbb{H}^{2}$ to
$\mathbb{H}^{N}$ and distinguise the thickness of simplexes of a compact
convex set $C$ (simplex) )$t_{g}(C)$ to the general thickness $t_{g}(C)$ and
the normal thickness $t_{N},$ which yields $t_{N}(C)\geq t_{g}(C).$

######  Lemma 10.

[16, Theorem 2, (2.3), pp. 54-55] Let $\Delta^{N}$ be an $N-$simplez in
$\mathbb{H}^{N}$ with edges $a_{ij}\in[s,\ell]\subset[\lambda_{N}(\ell),\ell$
where $\ell>0$ and $\ell_{N}(\ell)$ is given by

$\cosh\lambda_{N}(\ell)=\sqrt{\cosh^{2}\ell-2f_{N}(\cosh\ell-1)[1+\frac{N}{N+1}(\cosh\ell)-1,]}$

where

$f_{N})=\left\\{\begin{array}[]{ll}\frac{2}{N+1}&for\ odd\ N,\\\
\frac{2(N+1)}{N(N+2)}&for\ even\ N\par\end{array}\right.$

Then the general thickness $t_{g}$ of $\Delta^{N}$ satisfies:

$t_{g}\geq\cosh^{-1}(1+\frac{\cosh s-\cosh\lambda_{N}(\ell)}{\cosh^{2}\ell}).$

######  Theorem 16 (Isometric embedding of a dynamic Fermat-Steiner-Frechet
multitree to a Frechet almost regular multisimplex).

The embedding of a dynamic intermediate Fermat-Steiner Frechet multitree
$MS\to F\Delta^{N},$ $F\Delta_{\epsilon_{i}}^{N}$ is a family of almost
regular multiFrechet simplexes for
$s=\ell-\max\\{|\epsilon_{1}|,\cdots|\epsilon_{i}|\\}$ very close to $\ell$,
which gives a collection of isometries up to an additive constant, where the
implied constants only depends on $N$ and $\epsilon,$ such that
$t_{g}(F\Delta^{N})\geq\frac{1}{2\epsilon}\log 3$ for $s\leq a_{ij}\leq\ell$
and $(s,\ell)\in DW_{\mathbb{H}_{-\epsilon^{2}}^{N}}.$

###### Proof.

Let $A_{1}A_{2}\cdots A_{N+1}$ be an almost regular simplex for $s$ very close
to $\ell$ and $X_{1}$ is the corresponding Fermat point. By taking $X_{1}$ as
a reference point, the rays $X_{1}A_{i}$ isuuing from $X_{1}$ intersect
$\partial\mathbb{H}_{-\epsilon^{2}}^{N}$ at $A_{i}^{\circ},$ for
$i=1,2,\cdots,N,$ which yields that $A_{1}^{\circ}A_{2}^{\circ}\cdots
A_{N}^{\circ}$ is an ideal almost regular $N$simplex. By using Gromov’s
observation that we can assign a Fermat tree having a maximal length metric to
each $A_{1}^{\circ}A_{2}^{\circ}\cdots A_{N}^{\circ},$ which has an almost
maximal volume and by applying Theorem 15, we can embed isometrically up to an
additive constant to each member ($N$ almost regular simplex) of
multiFrechetsimplexes $F\Delta_{\epsilon_{i}}^{N},$ for
$s=\ell-\max\\{|\epsilon_{1}|,\cdots|\epsilon_{i}|\\}$ very close to $\ell:$
$s\leq a_{ij}\leq\ell.$

The Fermat point of $A_{1}A_{2}\cdots A_{N+1}$ and
$A_{1}^{\circ}A_{2}^{\circ}\cdots A_{N+1}^{\circ}$ remains the same, because

$\sum_{i}^{N+1}\frac{\exp_{X_{1}}^{-1}X_{1}A_{i}}{|X_{1}A_{i}|}=\sum_{i}^{N+1}\frac{\exp_{X_{1}}^{-1}X_{1}A_{i}^{\circ}}{|X_{1}A_{i}^{\circ}|}=0.$

∎

Let $S$ be an intermediate Fermat-Steiner tree with two intermediate Fermat-
Steiner points $X_{1}$ and $X_{2}$ inside the ideal $N$ simplex
$A_{1}^{\circ}\cdots A_{N+1}^{\circ}$ in $\mathbb{H}_{k}^{N}$ having degree
$p$ and $(N+2)-p$ and the boundary ideal vertices $A_{i}^{\circ}$ having
degree one, for $3\leq p\leq N-2.$

######  Lemma 11.

Convergence of $\Delta^{3}$ [24, Example, p. 234] If $X_{1}X_{2}\to\infty,$
for $N=3,$ $p=3,$ then $\Delta^{3}$ converges exponentially fast converges to
the union of two triangles spanned by the triples
$\\{A_{1}^{\circ},A_{2}^{\circ},\frac{X_{1}+X_{2}}{2}\\}$ and
$\\{A_{3}^{\circ},A_{4}^{\circ},\frac{X_{1}+X_{2}}{2}\\}.$

######  Lemma 12 (Extension of convergence of an $N$simplex to a $p$simplex
and $N+3-p$ simplex ).

If $X_{1}X_{2}\to\infty,$ then $\Delta^{N}$ exponentially fast converges to
the union of two hyperbolic simplexes spanned by the $p$-tuple
$\\{A_{1}^{\circ},A_{2}^{\circ},\cdots,A_{p-1}^{\circ},\frac{X_{1}+X_{2}}{2}\\}$
and the
$(N+3-p)$-tuple
$\\{A_{p}^{\circ},A_{p+1}^{\circ},\cdots,A_{N+2-p}^{\circ},\frac{X_{1}+X_{2}}{2}\\},$
respectively.

In [24, Example,p. 234], Gromov also observed that if the basic invariant of
$\Delta^{3}$ $X_{1}X_{2}<\infty,$ then the geometry of the ideal simplex
$\Delta^{3}$ is close to that of the regular ideal simplex.

In Nature, an evolutionary tree with respect to a boundary closed polyhedron
in $\mathbb{R}^{3}$ tends to maximize the volume of the closed polyhedron
formed by the boundary vertices with a mimimum communication via minimum
transfer of mass along the branches. The evolutionary tree reduces its length
by placing intermediate Fermat-Steiner points inside the boundary polyhedron.

######  Definition 19 (Degree of intelligence).

We call degree of intelligence of an evolutionary tree in $\mathbb{H}_{K}^{N}$
the number of intermediate Fermat Steiner points, which correspond to an
intermediate Fermat Steiner tree of an $N$-simplex in $\mathbb{H}_{K}^{N}.$

######  Definition 20 (Maximum degree of intelligence of an evolutionary
tree).

The maximum degree of intelligence of an evolutionary tree, which correspond
to an intermediate Fermat Steiner tree of an $N$-simplex in
$\mathbb{H}_{K}^{N}$ is $N-2.$

Consider two trees $T$ and $T^{\prime}$ having the same set of boundary
vertices in $\mathbb{H}_{k}^{2}.$

######  Definition 21 (Intelligent trees).

The tree $T$ is more intelligent than then tree $T^{\prime}$ if the degree of
intelligence of $T$ is greater than the degree of intelligence of
$T^{\prime}.$

We continue with Gromov’s definition of thin triangles in $\delta$ hyperbolic
metric spaces, which help us to view $\delta$ hyperbolic metric spaces as
metric trees with a prescribed thickness ([24],[10, 1.1 Fefinition, Fig.
H.1,p. 399]).

######  Definition 22.

Thin triangle,[24, p. 183][10, 1.1 Fefinition, Fig. H.1,p. 399] A geodesic
triangle in a metric space is said to be $\delta$ thin if each of its sides is
contained in the $\delta$ neighborhood of the union of the other two sides.

A metric (geodesic) tree with zero thickness is an $\mathbb{R}$ tree (Zero
hyperbolic geodesic space)

######  Definition 23 (Degree of intelligence of a $\delta$ metric tree).

The degree of intelligence of a $\delta$ metric tree embedded in an $N$
simplex is the degree of intelligence of the corresponding $zero$ ($\delta=0$)
metric tree (intermediate Fermat-Steiner tree) with respect to the same
boundary $N$-simplex.

Consider an intermediate Fermat-Steiner tree having two degrees of
intelligence $X_{1},$ $X_{2}$ with corresponding degrees $p$ and $N+2-p,$
respectively inside the ideal simplex $A_{1}^{\circ}A_{2}^{\circ}\cdots
A_{N}^{\circ}\in\mathbb{H}_{-\epsilon^{2}}^{N}.$

######  Theorem 17 (Reduction of the degree of intelligence of a metric
$\delta$ tree).

If the hyperbolic distance $X_{1}X_{2}\to infinity,$ then the degree of
intelligence of the $\frac{1}{2\epsilon^{2}}\log 3$ metric tree is reduced by
one degree from two to one.

###### Proof.

By applying Lemma 12 and taking into account $X_{1}X_{2}\to\infty,$
$\Delta^{N}$ exponentially fast converges to the union of two hyperbolic
simplexes spanned by the $p-tuple$
$\\{A_{1}^{\circ},A_{2}^{\circ},\cdots,A_{p-1}^{\circ},\frac{X_{1}+X_{2}}{2}\\}$
and the $N+3-p$tuple
$\\{A_{p}^{\circ},A_{p+1}^{\circ},\cdots,A_{N+2-p}^{\circ},\frac{X_{1}+X_{2}}{2}\\},$
respectively.

Hence, one may consider the vertex $\frac{X_{1}+X_{2}}{2}$ as the weighted
Fermat point with respect to the union of the two hyperbolic simplexes, which
minimize the objective function (zero Fermat (metric) geodesic tree)

$w_{1}A_{1}^{\circ}\frac{X_{1}+X_{2}}{2}+w_{2}A_{2}^{\circ}\frac{X_{1}+X_{2}}{2}+w_{p-1}A_{p-1}^{\circ}\frac{X_{1}+X_{2}}{2}+\cdots+w_{N+2-p}A_{N+2-p}^{\circ}\frac{X_{1}+X_{2}}{2}.$

Taking into account the corresponding $\delta=\frac{1}{\epsilon^{2}}\log 3$ we
may consider a $k\delta$ hyperbolic metric tree ([24, Lemma,p. 183]) as a
union of the $\delta$ branches $A_{i}^{\circ}\frac{X_{1}+X_{2}}{2},$ for
$i=1,2,\cdots,N+1,$ for a proper $N+1$ tuple of weights (positive real
numbers), we obtain a metric tree structure having one degree of intelligence
(one Fermat point). ∎

##  5\. A variational approach to the weighted Fermat-Frechet problem for a
given sextuple of edge lengths determining a $3-$simplex in the three-
dimensional $K-$Space

In this section, we obtain a method to differentiate the length of a geodesic
arc with respect to a geodesic arc in the 3 $K-$Space, which is different than
the method derived by Schlafli ([40])and Luo ([35]). This new variational
technique can be applied to solve the weighted Fermat-Frechet problem for
boundary incongruent tetrahedra in the 3 $K-$Space and determine the
corresponding weighted Fermat trees for each derived tetrahedron, such that
their edges belongs to the Dekster Wilker spherical domain
$DW_{\mathbb{S}_{\frac{1}{\sqrt{K}}}^{3}}(\ell,s)$ or hyperbolic domain
$DW_{\mathbb{H}_{K}^{3}}(\ell,s).$

Let $A_{1}A_{2}A_{3}A_{4}$ be a tetrahedron in the 3$K-$Space and $A_{0}$ be
the corresponding weighted Fermat point inside the tetrahedron.

We denote by $\alpha$ the dihedral angle formed by the planes $\triangle
A_{1}A_{2}A_{3}$ and $\triangle A_{1}A_{2}A_{0}$ and by $\alpha_{g_{4}}$ the
dihedral angle formed by the planes $\triangle A_{1}A_{2}A_{3}$ and $\triangle
A_{1}A_{2}A_{4},$ by $h_{0,12}$ the height of $\triangle A_{1}A_{2}A_{3}$ from
$A_{0}$ to $A_{1}A_{2},$

by $A_{0,12}$ the projection of $A_{0}$ to $A_{1}A_{2}$ by $A_{0,123}$ the
projection of $A_{0}$ to the $K-$plane defined by $\triangle A_{1}A_{2}A_{3}$
by $h_{0,123}$ the length of $A_{0}A_{123},$ by $x_{i}$ the length of
$A_{i}A_{123},$ by $d$ the length of $A_{0,12}A_{0,123}$ and by $l$ the length
of $A_{0,12}A_{2},$ for $i=1,2,3.$

We set $\alpha_{ijk}\equiv\angle A_{i}A_{j}A_{k},$ $\beta\equiv\angle
A_{1}A_{2}A_{0,123},$ $h_{0,12}\equiv A_{0}A_{0,12},$ $h_{0,123}\equiv
A_{0}A_{0,123}$ and $l\equiv A_{2}A_{0,12}.$

for $i,j,k=0,1,2,3$

######  Theorem 18.

The geodesic edges $a_{03}$ and $a_{04}$ can be expressed as functions of
$a_{01},a_{02},\alpha:$

$\displaystyle\cos\kappa a_{03}=\cos\kappa a_{02}\cos\kappa a_{23}+$
$\displaystyle+\sin\kappa a_{23}\cos\kappa h_{0,12}\cos\alpha_{123}\sin\kappa
l(a_{01},a_{02})+$ $\displaystyle+\sin\kappa a_{23}\sin\alpha_{123}\cos\alpha$
(5.1)

$\displaystyle\cos\kappa a_{04}=\cos\kappa a_{02}\cos\kappa a_{24}+$
$\displaystyle+\sin\kappa a_{24}\cos\kappa h_{0,12}\cos\alpha_{124}\sin\kappa
l(a_{01},a_{02})+$ $\displaystyle+\sin\kappa
a_{24}\sin\alpha_{124}\cos(\alpha_{g_{4}}-\alpha)$ (5.2)

###### Proof.

From the cosine law in $\triangle A_{1}A_{2}A_{3}\in K$-plane, we get:

$\displaystyle\cos\alpha_{102}=\frac{\cos\kappa a_{12}-\cos\kappa
a_{01}\cos\kappa a_{02}}{\sin\kappa a_{01}\sin\kappa a_{02}}$ (5.3)

From the sine law in $\triangle A_{2}A_{0}A_{0,12},$ we get:

$\displaystyle\sin\alpha_{120}=\frac{\sin\kappa h_{0,12}}{\sin\kappa a_{02}}$
(5.4)

By substituting (5.23)in the sine law of $\triangle A_{1}A_{2}A_{0},$ we
derive:

$\displaystyle\frac{\sin\kappa a_{12}}{\sin\alpha_{102}}=\frac{\sin\kappa
a_{1}}{\sin\alpha_{120}}=\frac{\sin\kappa a_{01}\sin\kappa a_{02}}{\sin\kappa
h_{0,12}}$ (5.5)

or

$\displaystyle\sin\kappa h_{0,12}=\frac{\sin\kappa a_{01}\sin\kappa
a_{02}}{\sin\kappa a_{12}}\sin\alpha_{102}.$ (5.6)

By substituting (5.3) in (5.6), we obtain:

$\displaystyle\sin\kappa h_{0,12}=\frac{\sin\kappa a_{01}\sin\kappa
a_{02}}{\sin\kappa a_{12}}\sqrt{1-(\frac{\cos\kappa a_{12}-\cos\kappa
a_{01}\cos\kappa a_{02}}{\sin\kappa a_{01}\sin\kappa a_{02}})^{2}}$ (5.7)

From the cosine law in $\triangle A_{0}A_{0,123}A_{3},$ we get:

$\displaystyle\cos\kappa a_{03}=\cos\kappa h_{0,123}\cos\kappa x_{3}$ (5.8)

From the cosine law in $\triangle A_{0,123}A_{3}A_{2},$ we have:

$\displaystyle\cos\kappa x_{3}=\cos\kappa x_{2}\cos\kappa a_{23}+\sin\kappa
x_{2}\sin\kappa a_{23}\cos(\alpha_{123}-\beta)$ (5.9)

By substituting (5.9) in (5.8), we get:

$\displaystyle\cos\kappa a_{03}=\cos\kappa x_{2}\cos\kappa h_{0,123}\cos\kappa
a_{23}+$ $\displaystyle+\sin\kappa x_{2}\cos\kappa h_{0,123}\sin\kappa
a_{23}\cos(\alpha_{123}-\beta)$ (5.10)

From the cosine law in $\triangle A_{0,123}A_{2}A_{0},$ $\triangle
A_{0}A_{0,12}A_{2},$ we get respectively:

$\cos\kappa a_{2}=\cos\kappa x_{2}\cos\kappa h_{0,123}$ (5.11)

and

$\cos\kappa h_{0,123}=\cos\kappa\frac{\cos\kappa h_{0,12}}{\cos\kappa d}.$
(5.12)

By substituting (5.11) and (5.12) in (5), we have:

$\displaystyle\cos\kappa a_{03}=\cos\kappa a_{02}\cos\kappa a_{23}+$
$\displaystyle+\sin\kappa a_{23}\sin\kappa x_{2}\frac{\cos\kappa
h_{0,12}}{\cos\kappa d}\cos\alpha_{123}\cos\beta+$ $\displaystyle+\sin\kappa
a_{23}\sin\kappa x_{2}\frac{\cos\kappa h_{0,12}}{\cos\kappa
d}\sin\alpha_{123}\sin\beta$ (5.13)

From the sine law and cosine law in $\triangle A_{0,123}A_{0,12}A_{2},$ we
get:

$\displaystyle\sin\kappa x_{2}=\frac{\sin\kappa d}{\sin\beta}$ (5.14)

$\displaystyle\cos\kappa x_{2}=\cos\kappa d\cos\kappa l.$ (5.15)

By substituting (5.14) and (5.15) in (5), we have:

$\displaystyle\cos\kappa a_{03}=\cos\kappa a_{02}\cos\kappa a_{23}+$
$\displaystyle+\sin\kappa a_{23}\tan\kappa d\cos\kappa
h_{0,12}\cos\alpha_{123}\cos\beta\sin\kappa x_{2}+$ $\displaystyle+\sin\kappa
a_{23}\tan\kappa d\cos\kappa h_{0,12}\sin\alpha_{123}\sin\beta\sin\kappa
x_{2}$ (5.16)

From the cosine law and sine law in $\triangle A_{0,123}A_{0,12}A_{2},$ we
get:

$\displaystyle\cos\beta=\frac{\cos\kappa d-\cos\kappa l\cos\kappa
x_{2}}{\sin\kappa l\sin\kappa x_{2}}$ (5.17)

$\displaystyle\sin\beta=\frac{\sin\kappa d}{\sin\kappa x_{2}}$ (5.18)

$\displaystyle\cos\kappa l=\frac{\cos\kappa a_{02}}{\cos\kappa h_{0,12}}$
(5.19)

Taking into account that $h_{012}=h_{012}(a_{01},a_{02}),$ (5.19) yields:

$l=l(a_{01},a_{02})$

By substituting (5.17) and (5.18) in (5), we have:

$\displaystyle\cos\kappa a_{03}=\cos\kappa a_{02}\cos\kappa a_{23}+$
$\displaystyle+\sin\kappa a_{23}\cos\kappa h_{0,12}\cos\alpha_{123}\sin\kappa
l+$ $\displaystyle+\sin\kappa a_{23}\cos\kappa
h_{0,12}\sin\alpha_{123}\tan\kappa d$ (5.20)

From the cosine law and sine law in $\triangle A_{0}A_{0,12}A_{0,123},$ we
get:

$\displaystyle\cos\kappa h_{0,123}=\frac{\cos\kappa h_{0,12}}{\cos\kappa d}$
(5.21)

$\displaystyle\sin\kappa h_{0,123}=\frac{\sin\kappa h_{0,12}}{\sin\alpha}$
(5.22)

$\displaystyle\cos\kappa h_{0,123}=\cos\kappa d\cos\kappa h_{0,12}+\sin\kappa
d\sin\kappa h_{0,12}\cos\alpha$ (5.23)

Substituting (5.21), (5.22) in (5.23) yields:

$\displaystyle\tan\kappa d=\tan\kappa h_{0,12}\cos\alpha$ (5.24)

By substituting (5.24) in (5), we obtain (18).

By working cyclically and changing the index $3\to 4,$ we derive (18).

∎

######  Proposition 4.

$\displaystyle\kappa a_{04}=\kappa
a_{04}(a_{01},a_{02},a_{03};a_{12},a_{13},a_{14},a_{21},a_{23},a_{24},a_{34},\kappa).$
(5.25)

###### Proof.

By solving (18) with respect to $\cos\alpha$ and substituting this variable in
(18) and by replacing $\alpha_{123},$ $\alpha_{124},$ from the cosine law and
the sine law in $\triangle A_{1}A_{2}A_{4}$ and $A_{1}A_{2}A_{3}$ in (18) we
get (5.25). ∎

######  Theorem 19 (The weighted Fermat-Torricelli Frechet solution).

The following equations depending on the variable edge lengths
$a_{01},a_{02},a_{03}$ the constant sectional curvature $K$ and the edge
lengths $\\{a_{ij}\\}$ provide a necessary condition for the location of the
weighted floating Fermat-Torricelli trees $\\{a_{01},a_{02},a_{03},a_{04}\\},$
which belong to the weighted Fermat-Frechet (multitree solution in the
3$K$-Space:

$\displaystyle\sum_{i=2}^{4}\frac{B_{i}}{\sinh Ka_{01}\sinh Ka_{0i}}(\cosh
Ka_{01}\cosh Ka_{0i}-\cosh Ka_{1i})-$ $\displaystyle\sum_{i=1,{i\neq
j}}^{4}\frac{B_{i}}{\sinh Ka_{0j}\sinh Ka_{0i}}(\cosh Ka_{0j}\cosh
Ka_{0i}-\cosh Ka_{ji})=0,$

$\operatorname{Vol}(A_{0},A_{i},A_{j},A_{k})<\operatorname{Vol}(C_{1}C_{2},C_{3},C_{4}).$
(5.26)

for $i,j,k=1,2,3,4,i\neq j\neq k.$

###### Proof.

It is a direct consequence of Proposition 4 and Theorems 6, 8 for $N=4.$ ∎

In [55, Problem 5, Theorem 3], we study the weighted Fermat-Steiner problem
for a boundary tetrahedron $A_{1}A_{2}A_{3}A_{4}$ in $\mathbb{R}^{3}.$ The
weighted Fermat-Steiner solution is a tree having two weighted Fermat
(Steiner) points $A_{0}$ and $A_{0}^{\prime}$ inside $A_{1}A_{2}A_{3}A_{4}.$
The weighted Fermat Steiner tree topology consists of the branches (line
segments)
$\\{A_{1}A_{0},A_{2}A_{0},A_{0}A_{0}^{\prime},A_{0}^{\prime}A_{3},A_{0}^{\prime}A_{4}\\},$
with corresponding weights
$\\{B_{1},B_{2},\frac{B_{0}+B_{0}^{\prime}}{2},B_{3},B_{4}\\}.$ We call degree
of intelligence of a a weighted Fermat-Steiner network with respect to a
boundary $N$-simplex in $\mathbb{R}^{N}$ the number of weighted Fermat-Steiner
points it possesses inside the simplex. Thus, a full weighted Fermat-Steiner
network for boundary tetrahedra having two weighted Fermat-Steiner points has
two degrees of intelligence.

In [2, Figures 8, 9,pp. 18-19], P. Alexandrov makes an elegant exposition of
algebraic complexes by considering them as higher dimensional generalization
of ordinary directed polygonal paths, taking into account that a line which is
traversed twice in opposite directions does not count. By applying this
technique to a weighted Fermat-Steiner network for boundary tetrahedra, we may
isolate the two degrees of intelligence in two non-intersecting triangles
$\triangle A_{1}A_{2}A_{0}$ and $\triangle A_{1}A_{2}A_{0}^{\prime}.$

We consider the following two networks:

$\bullet$ a weighted Fermat-Steiner tree for $A_{1}A_{2}A_{3}A_{4},$ such
that:

$f(A_{0},A_{0}^{\prime})=B_{1}A_{1}A_{0}+B_{2}A_{2}A_{0}+B_{3}A_{3}A_{0}^{\prime}+B_{4}A_{4}A_{0}^{\prime}+\frac{B_{0}+B_{0}^{\prime}}{2}A_{0}A_{0}^{\prime}\to
min$

$\bullet$ the two broken lines formed by the boundaries $\triangle
A_{1}A_{2}A_{0}$ and $A_{3}A_{4}A_{0}^{\prime}$ connected by their minimum
distance $A_{0}A_{0}^{\prime}$

By using the orientation

$A_{0}\to A_{1}\to A_{2}\to A_{0}\to A_{0}^{\prime}\to A_{3}\to A_{4}\to
A_{0}^{\prime}\to A_{0},$

we obtain the following proposition, which deals with isolated intelligence of
the boundaries $\triangle A_{1}A_{2}A_{0}$ and $A_{3}A_{4}A_{0}^{\prime}.$

######  Proposition 5 (A directed weighted Fermat-Steiner Frechet isolated
multitree for Frechet tetrahedra in $\mathbb{R}^{3}$ in the sense of P.
Alexandrov).

If the orientation

$A_{0}\to A_{1}\to A_{2}\to A_{0}\to A_{0}^{\prime}\to A_{3}\to A_{4}\to
A_{0}^{\prime}\to A_{0},$

for the boundary triangles $\triangle A_{1}A_{2}A_{0}$ and
$A_{3}A_{4}A_{0}^{\prime}$ occurs for incogruent boundary tetrahedra derived
by a sextuple of positive real numbers under the conditions of Blumenthal,
Herzog, Dekster-Wilker, such that
$\\{A_{1}A_{0},A_{2}A_{0},A_{0}A_{0}^{\prime},A_{0}^{\prime}A_{3},A_{0}^{\prime}A_{4}\\}$
is a union of weighted Fermat-Steiner trees (Fermat-Steiner Frechet multitree)
with corresponding weights
$\\{B_{1},B_{2},\frac{B_{0}+B_{0}^{\prime}}{2},B_{3},B_{4}\\}$ for the
boundary tetrahedron $A_{1}A_{2}A_{3}A_{4},$ then we derive two boundary
triangles with two isolated degrees of intelligence for all incogruent
boundary tetrahedra (Frechet multitetrahedron).

###### Proof.

The orientation

$A_{0}\to A_{1}\to A_{2}\to A_{0}\to A_{0}^{\prime}\to A_{3}\to A_{4}\to
A_{0}^{\prime}\to A_{0}$

in the sense of P. ALexandrov cancels the line segment $A_{0}$ and
$A_{0}^{\prime}$ from the weighted Fermat-Steiner tree
$\\{A_{1}A_{0},A_{2}A_{0},A_{0}A_{0}^{\prime},A_{0}^{\prime}A_{3},A_{0}^{\prime}A_{4}\\},$
which leads to two boundaries $\triangle A_{1}A_{2}A_{0}$ and
$A_{3}A_{4}A_{0}^{\prime},$ with two isolated degrees of intelligence at the
vertices $A_{0}$ and $A_{0}^{\prime}.$ Therefore, taking into account the
conditions for a sextuple of positive real numbers determining the edge
lengths of a maximum of thirty incongruent tetrahedra (Frechet
multitetrahedron) studied by Blumenthal, Herzog and Dekster Wilker, we may get
a union of isolated boundary triangles, which correspond to the weighted
Fermat-Steiner Frechet tree (multitree) for the Frechet multitetrahedron. ∎

In [17] and [18], Edelstein, Schwartz and Eremenko proved Gehrink’s problem on
linked curves in $\mathbb{R}^{3}.$

######  Lemma 13.

[17], [18] If $C_{1}$ and $C_{2}$ are two linked closed curves (any continuous
deformation of $C_{1}$ to a point intersect $C_{2}$) in $\mathbb{R}^{3}$ and
the distance between $C_{1}$ and $C_{2}$ is 1, then the length of $C_{1}$ or
$C_{2}$ is at least $2\pi.$

######  Proposition 6 (Gehring’s linked curved inequality associated with the
weighted Fermat-Steiner Frechet multitree in $\mathbb{R}^{3}$).

If $A_{1}A_{2}A_{3}A_{4}$ belongs to a union of incongruent tetrahedra, which
give the Frechet multitetrahedron, formed by a given sextuple of positive real
numbers, determining edge lengths, $C_{1}\equiv boundary(\triangle
A_{1}A_{0}A_{0}^{\prime}),$ $C_{2}\equiv boundary(\triangle A_{2}A_{3}A_{4}),$
$A_{0},$ $A_{0}^{\prime}$ the corresponding Fermat-Steiner points and
$dist(\triangle A_{1}A_{0}A_{0}^{\prime},\triangle A_{2}A_{3}A_{4}))=1,$ then
the length of $boundary(\triangle A_{1}A_{0}A_{0}^{\prime})$ or
$boundary(\triangle A_{2}A_{3}A_{4})$ is at least $2\pi.$

###### Proof.

By applying lemma!13, for $C_{1}\equiv boundary(\triangle
A_{1}A_{0}A_{0}^{\prime}),$ $C_{2}\equiv boundary(\triangle A_{2}A_{3}A_{4}),$
we derive that the length of $boundary(\triangle A_{1}A_{0}A_{0}^{\prime})$ or
$boundary(\triangle A_{2}A_{3}A_{4})$ is at least $2\pi.$ ∎

######  Remark 3.

A generalization of Proposition 6 for Steiner trees in $\mathbb{R}^{3}$ may
give a new perspective to the Steiner ratio conjecture, which was proved for
manifolds by Ivanov-Tuzhlin-Cieslik in [31].

##  6\. Some computations on the variation of Fermat trees for tetrahedra
having one or three vertices at infinity in $\mathbb{R}^{3}$

In this section, we present a new class of weighted Fermat trees for a
tetrahedron $A_{1}A_{2}A_{3}A_{4}$ having the vertex $A_{4}$ at infinity or
the three vertices $A_{1},$ $A_{2},$ $A_{3}$ at infinity, such that $A_{4}$
belongs to the perpendicular line w.r to the plane defined by $\triangle
A_{1}A_{2}A_{3}$ at the corresponding weighted Fermat point $A_{0,123}$ of
$\triangle A_{1}A_{2}A_{3},$ which is derived by setting $B_{4}=0$ in the
weighted Problem for $A_{1}A_{2}A_{3}A_{4}$ and the length of $A_{4}A_{0,123}$
is a large positive real number $M,$ for $B_{4}=1,$ and $B_{i}=B_{i}(M),$ are
positive linear function w.r. to $M,$ for $i=1,2,3.$

Let $A_{1}A_{2}A_{3}A_{4}$ be a tetrahedron having the vertex $A_{4}$ at
infinity in $\mathbb{R}^{3},$ with corresponding weights
$B_{1}(M)=b_{1}M+c_{1}>0,$ $B_{2}(M)=b_{2}M+c_{2}>0,$ and
$B_{3}(M)=b_{3}M+c_{3}>0,$ $B_{4}=1$ having positive real values for a given
large number $M$ and given real numbers $b_{i},$ $c_{i}$ for $i=1,2,3.$

We denote by $A_{0,123}$ the corresponding weighted Fermat-Torricelli point of
$\triangle A_{1}A_{2}A_{3}$ for given weights $B_{1}(M),$ $B_{2}(M)$ and
$B_{3}(M),$ which satisfy the inequalities (2.1) of the weighted floating case
of Theorem 1 by setting $B_{4}=0.$

We set $\varphi\equiv\angle A_{0,123}A_{1}A_{3},$
$a_{i,0123}\equiv\|A_{0,123}A_{i}\|$ and $B_{i}\equiv B_{i}(M),$ for
$i=1,2,3.$

######  Lemma 14.

[50] The exact position of the weighted Fermat-Torricelli tree w.r to
$\triangle A_{1}A_{2}A_{3}$ is given by:

$\varphi=\operatorname{arccot}\left(\frac{\sin(\alpha_{213})-\cos(\alpha_{213})\cot(\arccos{\frac{B_{3}^{2}-B_{1}^{2}-B_{2}^{2}}{2B_{1}B_{2}}})-\frac{a_{13}}{a_{12}}\cot(\arccos{\frac{B_{2}^{2}-B_{1}^{2}-B_{3}^{2}}{2B_{1}B_{3}}})}{-\cos(\alpha_{213})-\sin(\alpha_{213})\cot(\arccos{\frac{B_{3}^{2}-B_{1}^{2}-B_{2}^{2}}{2B_{1}B_{2}}})+\frac{a_{13}}{a_{12}}}\right)$
(6.1)

and

$a_{1,0123}=\frac{\sin\left(\varphi+\arccos{\frac{B_{2}^{2}-B_{1}^{2}-B_{3}^{2}}{2B_{1}B_{3}}}\right)a_{13}}{\sin\left(\arccos{\frac{B_{2}^{2}-B_{1}^{2}-B_{3}^{2}}{2B_{1}B_{3}}}\right)}.$
(6.2)

From Lemma 14, we derive that:

######  Lemma 15.

The line segments $a_{2,0123}$ and $a_{3,0123}$ depend on $B_{1},$ $B_{2},$
$B_{3},$ $a_{12},$ $a_{13},$ $a_{23}$ and $\varphi:$

$a_{2,0123}=\sqrt{a_{1,0123}^{2}+a_{12}^{2}-2a_{1,0123}a_{12}\cos(\angle
A_{2}A_{1}A_{3}-\varphi)}$ (6.3)

and

$a_{3,0123}=\sqrt{a_{1,0123}^{2}+a_{13}^{2}-2a_{1,0123}a_{13}\cos(\varphi)}.$
(6.4)

We assume that $A_{4}$ lies on the normal line w.r. to the plane defined by
$\triangle A_{1}A_{2}A_{3},$ for $B_{4}=1$ and the the length of
$A_{0,123}A_{4}$ is $M.$

We recall that the Cayley-Menger determinant $D(S)$ is given by:

$D(S)=\operatorname{det}\left(\begin{array}[]{ccccc}0&a_{12}^{2}&a_{13}^{2}&a_{14}^{2}&1\\\
a_{12}^{2}&0&a_{23}^{2}&a_{24}^{2}&1\\\
a_{13}^{2}&a_{23}^{2}&0&a_{34}^{2}&1\\\
a_{14}^{2}&a_{24}^{2}&a_{34}^{2}&0&1\\\ 1&1&1&1&0\\\ \end{array}\right).$
(6.5)

We recall that $a_{ij}$ is length of the line segment $A_{i}A_{j},$
$\alpha_{ijk}\equiv\angle A_{i}A_{j}A_{k},$ the dihedral angle $\alpha$ is
defined by the planes formed by $\triangle A_{0}A_{1}A_{2}$ and $\triangle
A_{1}A_{2}A_{3},$ the dihedral angle $\alpha_{g_{4}}$ is defined by the planes
formed by $\triangle A_{1}A_{2}A_{3}$ and $\triangle A_{1}A_{2}A_{4},$
$h_{0,12}$ is the height of $\triangle A_{0}A_{1}A_{2}$ from $A_{0},$ by
$h_{0,12m}$ the distance of $A_{0}$ from the plane defined by $\triangle
A_{1}A_{2}A_{m},$ for $i,j,k=0,1,2,3,4$ and $m=3,4.$

The variable line segments $a_{03}(a_{01},a_{02},\alpha),$
$a_{04}(a_{01},a_{02},\alpha),$ are derived in [49, Formulas (2.14), (2.20) p.
116].

######  Proposition 7.

The variable lengths $a_{03}(a_{01},a_{02},\alpha),$
$a_{04}(a_{01},a_{02},\alpha;M),$ are given by:

$a_{03}(a_{01},a_{02},\alpha)=\sqrt{a_{02}^{2}+a_{23}^{2}-2a_{23}[\sqrt{a_{02}^{2}-h_{0,12}^{2}}\cos\alpha_{123}+h_{0,12}\sin\alpha_{123}\cos\alpha]}$
(6.6)

and

$a_{04}(a_{01},a_{02},\alpha;M)=\sqrt{a_{02}^{2}+a_{24}^{2}-2a_{24}[\sqrt{a_{02}^{2}-h_{0,12}^{2}}\cos\alpha_{124}+h_{0,12}\sin\alpha_{124}\cos(\alpha_{g_{4}}-\alpha)]},$
(6.7)

where

$h_{0,12}=h_{0,12}(a_{01},a_{02},a_{12})=\frac{a_{01}a_{02}}{a_{12}}\sqrt{1-\left(\frac{a_{01}^{2}+a_{02}^{2}-a_{12}^{2}}{2a_{01}a_{02}}\right)^{2}}.$
(6.8)

such that:

$\lim_{M\to+\infty}a_{04}(a_{01},a_{02},\alpha;M)=+\infty.$ (6.9)

###### Proof.

The angle $\alpha_{123}$ does not depend on $M:$

$\cos\alpha_{123}=\frac{a_{12}^{2}+a_{23}^{2}-a_{13}^{2}}{2a_{12}a_{23}},$
(6.10)

and

$\sin\alpha_{123}=\frac{\sqrt{(a_{12}+a_{23}+a_{13})(a_{23}+a_{13}-a_{12})(a_{12}+a_{13}-a_{23})(a_{12}+a_{23}-a_{13})}}{2a_{12}a_{23}}$
(6.11)

By replacing (6.10), (6.11) and (6.8) in (6.6) we derive that
$a_{03}=a_{03}(a_{01},a_{02},\alpha).$

We shall show that $a_{41},$ $a_{42},$ $\alpha_{124}$ depend on $M.$

From the right triangles $\triangle A_{1}A_{0,123}A_{4},$ $\triangle
A_{2}A_{0,123}A_{4}$ and taking into account (6.2) and (6.3), we get:

$a_{41}=a_{41}(M)=\sqrt{M^{2}+a_{1,0123}^{2}}$ (6.12)

and

$a_{42}=a_{42}(M)=\sqrt{M^{2}+a_{2,0123}^{2}}.$ (6.13)

where

$a_{4}^{2}=a_{2}^{2}+a_{24}^{2}-2a_{24}[\sqrt{a_{2}^{2}-h_{0,12}^{2}}\cos\alpha_{124}+h_{0,12}\sin\alpha_{124}(\cos\alpha_{g_{4}}\cos\alpha+\sin\alpha_{g_{4}}\sin\alpha)]$
(6.14)

where

$\cos\alpha_{124}=\frac{a_{12}^{2}+a_{24}^{2}-a_{14}^{2}}{2a_{12}a_{24}},$
(6.15)

$\sin\alpha_{124}=\frac{\sqrt{(a_{12}+a_{24}+a_{14})(a_{24}+a_{14}-a_{12})(a_{12}+a_{14}-a_{24})(a_{12}+a_{24}-a_{14})}}{2a_{12}a_{24}},$
(6.16)

$\alpha_{g_{4}}=\arccos\left(\frac{\left(\frac{a_{42}^{2}+a_{23}^{2}-a_{43}^{2}}{2a_{23}}\right)-\sqrt{a_{42}^{2}-h_{4,12}^{2}}\cos\alpha_{123}}{h_{4,12}\sin\alpha_{123}}\right)$
(6.17)

and

$h_{4,12}=h_{4,12}(a_{41},a_{42},a_{12})=\frac{a_{41}a_{42}}{a_{12}}\sqrt{1-\left(\frac{a_{41}^{2}+a_{42}^{2}-a_{12}^{2}}{2a_{41}a_{42}}\right)^{2}}.$
(6.18)

The dihedral angle $\alpha_{g_{4}}$ is derived by setting in (6.6) and (6.8)
the index from $0\to 4.$

By replacing (6.12), (6.13), (6.15), (6.16), (6.17), (6.8) in (6.7)

we derive that: $a_{04}=a_{04}(a_{01},a_{02},\alpha;M).$

∎

The position of the weighted Fermat-Torricelli tree of a tetrahedron
$A_{1}A_{2}A_{3}A_{4}$ is given by computing the volumes of the tetrahedra
$\operatorname{Vol}(A_{0}A_{i}A_{j}A_{k})$ for $i,j,k=1,2,3,4,$ via the Caley-
Menger determinant ([48, pp. 249-255]), which depend on $a_{1},$ $a_{2}$
,$\alpha$ and $M.$

We set
$C\equiv\frac{\sum_{i=1}^{4}\frac{B_{i}}{a_{i}}}{\operatorname{Vol}(A_{1}A_{2}A_{3}A_{4})}.$

######  Theorem 20.

The following four equations provide a necessary condition to determine the
position of the weighted Fermat-Torricelli tree at the interior of
$A_{1}A_{2}A_{3}A_{4}:$

$\left(\frac{B_{3}}{a_{3}\operatorname{Vol}(A_{0}A_{1}A_{2}A_{4})}\right)^{2}=\left(\frac{B_{4}}{a_{4}\operatorname{Vol}(A_{0}A_{1}A_{2}A_{3})}\right)^{2}=C^{2},$
(6.19)

$\left(\frac{B_{3}}{a_{3}\operatorname{Vol}(A_{0}A_{1}A_{2}A_{4})}\right)^{2}=\left(\frac{B_{1}}{a_{1}\operatorname{Vol}(A_{0}A_{2}A_{3}A_{4})}\right)^{2}=C^{2},$
(6.20)

$\left(\frac{B_{3}}{a_{3}\operatorname{Vol}(A_{0}A_{1}A_{2}A_{4})}\right)^{2}=\left(\frac{B_{2}}{a_{2}\operatorname{Vol}(A_{0}A_{1}A_{3}A_{4})}\right)^{2}=C^{2},$
(6.21)

and

$\left(\frac{B_{1}}{a_{1}\operatorname{Vol}(A_{0}A_{2}A_{3}A_{4})}\right)^{2}=\left(\frac{B_{2}}{a_{2}\operatorname{Vol}(A_{0}A_{1}A_{3}A_{4})}\right)^{2}=C^{2}.$
(6.22)

where

$B_{i}(M)=b_{i}M+c_{i}>0,$ (6.23)

for $i=1,2,3$

and

$B_{4}=1.$

###### Proof.

The objective function is given by:

$f(a_{1},a_{2},\alpha;M)=B_{1}a_{1}+B_{2}a_{2}+B_{3}a_{3}(a_{1},a_{2},\alpha)+B_{4}a_{4}(a_{1},a_{2},\alpha;M).$
(6.24)

or

$f(a_{1},a_{4},\alpha^{\prime};M)=B_{1}a_{1}+B_{4}a_{4}+B_{2}a_{2}(a_{1},a_{4},\alpha^{\prime})+B_{3}a_{3}(a_{1},a_{4},\alpha^{\prime};M).$
(6.25)

or

$f(a_{2},a_{3},\alpha^{\prime\prime};M)=B_{2}a_{2}+B_{3}a_{3}+B_{1}a_{1}(a_{2},a_{3},\alpha^{\prime\prime})+B_{4}a_{4}(a_{2},a_{3},\alpha^{\prime\prime};M).$
(6.26)

where $\alpha^{\prime}$ is the dihedral angle formed by $\triangle
A_{1}A_{0}A_{4}$ and $\triangle A_{1}A_{4}A_{2}$ and $\alpha^{\prime\prime}$
is the dihedral angle formed by $\triangle A_{2}A_{0}A_{3}$ and $\triangle
A_{2}A_{1}A_{3}.$

By differentiating (6.24) w.r to $\alpha,$ (6.25) w.r. to $\alpha^{\prime}$
and (6.26) w.r. to $\alpha^{\prime\prime}$ we derive (see also in [49, Formula
(2.25),pp. 117]):

$\displaystyle\frac{B_{3}}{a_{3}\operatorname{Vol}(A_{0}A_{1}A_{2}A_{4})}=\frac{B_{4}}{a_{4}\operatorname{Vol}(A_{0}A_{1}A_{2}A_{3})}={}$
$\displaystyle{}=\frac{B_{1}}{a_{1}\operatorname{Vol}(A_{0}A_{2}A_{3}A_{4})}=\frac{B_{2}}{a_{2}\operatorname{Vol}(A_{0}A_{1}A_{3}A_{4})}=C,$
(6.27)

The volume $\operatorname{Vol}(A_{0}A_{i}A_{j}A_{k}),$ for $i,j,k=1,2,3,4$ is
given by([48, pp. 249-255]):

$288\operatorname{Vol}(A_{0}A_{1}A_{2}A_{3})^{2}=D(\\{a_{1},a_{2},a_{3},a_{23},a_{13},a_{12}\\})$
(6.28)

$288\operatorname{Vol}(A_{0}A_{2}A_{3}A_{4})^{2}=D(\\{a_{4},a_{2},a_{3},a_{23},a_{43},a_{42}\\})$
(6.29)

$288\operatorname{Vol}(A_{0}A_{1}A_{3}A_{4})^{2}=D(\\{a_{1},a_{4},a_{3},a_{43},a_{13},a_{14}\\})$
(6.30)

and

$288\operatorname{Vol}(A_{0}A_{1}A_{2}A_{4})^{2}=D(\\{a_{1},a_{2},a_{4},a_{24},a_{14},a_{12}\\}).$
(6.31)

By squaring both parts of the equations in (6) and then by replacing (6.28),
(6.31), (6.30), (6.29) in the derived equations, we deduce (6.19), (6.20),
(6.21) and (6.22) which depend on $a_{1},$ $a_{2},$ $\alpha$ and $M.$

∎

For $M\to+\infty,$ the solution of the weighted Fermat-Torricelli problem is a
weighted Fermat-Torricelli tree with branches $A_{1}A_{0,123},$
$A_{2}A_{0,123},$ $A_{3}A_{0,123}$ and $A_{4}A_{0,123}.$ We call the weighted
Fermat-Torricelli tree for a tetrahedron having one vertex at infinity a large
tree because one of the four branches $A_{4}A_{0,123}\to\infty.$

The unique solution of the inverse problem for tetrahedra in $\mathbb{R}^{3}$
has been established in [49].

######  Problem 2.

Inverse weighted Fermat problem for tetrahedra in $\mathbb{R}^{3},$[49]

Given a point $A_{0}$ and a positive real number $C$ which belongs to the
interior of $A_{1}A_{2}A_{3}A_{4}$ in $\mathbb{R}^{3}$, does there exist a
unique set of positive weights $B_{i},$ such that

$B_{1}+B_{2}+B_{3}+B_{4}=C,$

for which $A_{0}$ minimizes

$f(A_{0})=\sum_{i=1}^{4}B_{i}a_{0i}.$

We denote by $\alpha_{i,j0k}$ the angle that is formulated by the line segment
$A_{0}A_{i}$ and the line segment that connects $A_{0}$ with the trace of the
orthogonal projection of $A_{i}$ to the plane defined by $\triangle
A_{j}A_{0}A_{k}.$ A positive answer w.r. to the inverse problem for
$A_{1}A_{2}A_{3}A_{4}$ is given in [49, Proposition 1]):

######  Lemma 16.

[49, Proposition 1, Solution of Problem 2] The weight $B_{i}$ are uniquely
determined by the formula:

$B_{i}=\frac{C}{1+\|\frac{\sin{\alpha_{i,k0l}}}{\sin{\alpha_{j,k0l}}}\|+\|\frac{\sin{\alpha_{i,j0l}}}{\sin{\alpha_{k,j0l}}}\|+\|\frac{\sin{\alpha_{i,k0j}}}{\sin{\alpha_{l,k0j}}}\|},$
(6.32)

where

$\frac{B_{j}}{B_{i}}=\frac{\sin{\alpha_{i,k0l}}}{\sin{\alpha_{j,k0l}}}$ (6.33)

$\frac{\sin{\alpha_{i,k0l}}}{\sin{\alpha_{j,k0l}}}=\sqrt{\|\frac{\sin^{2}\alpha_{k0m}-\cos^{2}\alpha_{m0i}-\cos^{2}\alpha_{k0i}+2\cos\alpha_{k0m}\cos\alpha_{m0i}\cos\alpha_{k0i}}{\sin^{2}\alpha_{k0m}-\cos^{2}\alpha_{m0j}-\cos^{2}\alpha_{k0j}+2\cos\alpha_{k0m}\cos\alpha_{m0j}\cos\alpha_{k0j}}\|}$
(6.34)

for $i,j,k,l=1,2,3,4$ and $i\neq j\neq k\neq l.$

For $B_{4}=0,$ we obtain the inverse weighted Fermat-Torricelli problem for
$\triangle A_{1}A_{2}A_{3}.$

######  Lemma 17.

[25] The weight $B_{i}$ are uniquely determined by the formula:

$B_{i}=\frac{C}{1+\frac{\sin{\alpha_{i0k}}}{\sin{\alpha_{j0k}}}+\frac{\sin{\alpha_{i0j}}}{\sin{\alpha_{j0k}}}},$
(6.35)

where

$\frac{B_{j}}{B_{i}}=\frac{\sin{\alpha_{i0k}}}{\sin{\alpha_{j0k}}}.$ (6.36)

for $i,j,k=1,2,3.$

######  Theorem 21.

If $M\to+\infty,$ for $B_{4}=1$ the solution of the inverse problem for the
tetrahedron $A_{1}A_{2}A_{3}A_{4}$ having the vertex $A_{4}$ at infinity in
$\mathbb{R}^{3}$ coincides with the solution of the inverse problem for
$\triangle A_{1}A_{2}A_{3}.$

###### Proof.

For $M\to+\infty,$ $A_{0}\to A_{0,123}$ and $\angle
A_{4}A_{0}A_{i}=90^{\circ},$ for $i=1,2,3.$ By replacing $\angle
A_{4}A_{0}A_{1}=\angle A_{4}A_{0}A_{2}=\angle A_{4}A_{0}A_{3}=90^{\circ}$ in
(6.34) and (6.36) for $i,j=1,2,3,$ $k,l,m=1,2,3,4$ and $i\neq j\neq k\neq l,$
we derive $B_{i}$

$\frac{B_{j}}{B_{i}}=\frac{\sin{\alpha_{i0k}}}{\sin{\alpha_{j0k}}},$

for $i,j,k=1,2,3$ and $B_{4}=1.$

∎

A direct consequence of Theorem 4, for

$B_{1}(M)+B_{2}(M)+B_{3}(M)+B_{4}(M)=C$

($C$ is a constant real number independent of $M$)

gives the following Proposition (the weights $B_{i}$ are independent of $M$)

######  Proposition 8.

For $M\to+\infty$ and

$B_{i}(M)=b_{i}M+c_{i}>0,$

$B_{4}=1,$

for $i=1,2,3$ such that

$b_{1}+b_{2}+b_{3}=0,$

$B_{i}$ are uniquely determined by the formula:

$B_{i}=\frac{C}{1+\frac{\sin{\alpha_{i0k}}}{\sin{\alpha_{j0k}}}+\frac{\sin{\alpha_{i0j}}}{\sin{\alpha_{j0k}}}},$
(6.37)

for $C=1+c_{1}+c_{2}+c_{3}$ and $i,j,k=1,2,3.$

As a future work, we may focus on studying properties of the Betti group of a
Frechet multisimplex.

## References

  * [1] M. Abramowitz and I.A. Stegun, _Handbook of Mathematical functions with Forumlas, Graphs and Mathematical Tables_ , Dover New York, (1972).
  * [2] P. Alexandroff, _Elementary concepts of topology._ New York: Dover Publications, (1961).
  * [3] I.D. Berg, I. G. Nikolaev, _On a extremal property of quadrilaterals in an Aleksandrov space of curvature $\leq K.$_ Burenkov, V. I. (ed.) et al., The interaction of analysis and geometry. International school-conference on analysis and geometry, Novosibirsk, Russia, August 23–September 3, 2004. Providence, RI: American Mathematical Society (AMS). Contemporary Mathematics 424 (2007), 1-15.
  * [4] L.M. Blumenthal and G.A. Garrett, _Characterization of spherical and pseudo-spherical sets of points_ Am. J. Math. 55, (1933), 619-640.
  * [5] L. M. Blumenthal,_A Budget of Curiosa Metrica_ , Amer. Math. Monthly, 66 (6), (1959), 453-460.
  * [6] V. Boltyanski, H. Martini, V. Soltan, _Geometric Methods and Optimization Problems_ , Kluwer, Dordrecht-Boston-London, 1999.
  * [7] K. Böröczky, _On an extremum property of the regular simplex in $S^{d}.$_ Intuitive geometry, Pap. Int. Conf., Siófok/Hung. 1985, Colloq. Math. Soc. János Bolyai 48 (1987), 117–121 (1987).
  * [8] B. H. Bowditch, _Notes on Gromov’s hyperbolicity criterion for path-metric spaces._ Ghys, É. (ed.) et al., Group theory from a geometrical viewpoint. Proceedings of a workshop, held at the International Centre for Theoretical Physics in Trieste, Italy, 26 March to 6 April 1990. Singapore: World Scientific. 64–167 (1991).
  * [9] B. H. Bowditch, _A course on geometric group theory._ MSJ Memoirs 16. Tokyo: Mathematical Society of Japan (MSJ), (2006).
  * [10] M. Bridson, A. Haefliger, _Metric spaces of non-positive curvature._ Grundlehren der Mathematischen Wissenschaften. 319. Berlin: Springer, (1999).
  * [11] I. Chavel, _Riemannian geometry: a modern introduction._ Cambridge University Press, Cambridge (1993).
  * [12] B.V. Dekster, J.B. Wilker, _Edge lengths guaranteed to form a simplex._ Arch. Math. 49 (1987), 351-366.
  * [13] B.V. Dekster and J.B. Wilker, _Large spherical simplexes._ J. Geom. 42, No. 1-2 (1991), 59-92.
  * [14] B.V. Dekster, J.B. Wilker, _Simplexes in spaces of constant curvature._ Geom. Dedicata 38 (1) (1991), 1-12.
  * [15] B.V. Dekster, _Spherical simplexes too big to exist._ J. Geom. 45 (1992), No. 1-2, 8-22.
  * [16] B.V. Dekster, _Thickness of a simplex whose edge lengths fall within a prescribed range._ Geom. Dedicata 68 (1997), No. 1 , 49–59.
  * [17] M. Edelstein, B. Schwatz, _On the length of linked curves_ , Israel J. Math. 23 (1976), No. 1, 94–95.
  * [18] A. Eremenko, _Gehring’s problem on linked curves_ , preprint 2009, https://www.math.purdue.edu/ eremenko/dvi/gehring.pdf.
  * [19] P. Fletcher, S. Venkatasubramanian, S. Joshi, _The geometric median on Riemannian manifolds with application to robust atlas estimation._ NeuroImage 45 (2009) S143-S152.
  * [20] M. Fréchet, _Sur la définition axiomatique d’une classe d’espaces vectoriels distancies applicables vectoriellement sur l’espace de Hilbert._ Ann. Math. (2) 36 (1935), 705-718.
  * [21] F. Herzog,_Completely Tetrahedral Sextuples_ , Amer. Math. Monthly, 66 (6), (1959), 460-464
  * [22] E.N. Gilbert, H.O. Pollak, _Steiner minimal trees._ SIAM J. Appl. Math. 16 (1968), 1–29.
  * [23] K. Gödel, _Über die metrische Einbettbarkeit der Quadrupel des $R^{3}$ in Kugelflächen_ Ergebnisse math. Kolloquium Wien 4 (1933), 16-17.
  * [24] M. Gromov, _Hyperbolic groups._ Essays in group theory, Publ., Math. Sci. Res. Inst. 8 (1987), 75-263.
  * [25] S. Gueron and R. Tessler, _The Fermat-Steiner problem_ , Amer. Math. Monthly, 109, (2002), 443–451
  * [26] U. Haagerup, Hans J. Munkholm, _Simplices of maximal volume in hyperbolic n-space._ Acta Math. 147 (1981), 1-11.
  * [27] H. Hadwiger, _Vorlesungen über Inhalt, Oberfläche und Isoperimetrie._ (German) Die Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, (1957).
  * [28] A. O. Ivanov and A. A. Tuzhilin, _Weighted minimal binary trees_. (Russian) Uspekhi Mat. Nauk 50 (1995), no. 3(303), 155–156; translation in Russian Math. Surveys 50 (1995), no. 3, 623–624.
  * [29] A.O. Ivanov, A.A. Tuzhilin, _Differential calculus on the space of Steiner minimal trees in Riemannian manifolds._ Sb. Math. 192, (6) (2001), 823-841 ; translation from Mat. Sb. 192 (6) (2001), 31-50.
  * [30] A.O. Ivanov and A.A. Tuzhilin, _Branching solutions to one-dimensional variational problems._ Singapore: World Scientific 2001.
  * [31] A.O. Ivanov, A.A. Tuzhilin, D. Cieslik, _The Steiner ratio for manifolds._ Math. Notes 74, No. 3 (2003), 367-374.
  * [32] A.O. Ivanov, A. Tuzhilin, Analytic deformations of minimal networks. J. Math. Sci., New York 248, No. 5, 621-635 (2020);translation from Fundam. Prikl. Mat. (2016) 21, No. 5, 159-180.
  * [33] H. Karcher,_Riemannian center of mass and mollifier smoothing._ Commun. Pure Appl. Math. 30 (1977), 509-541.
  * [34] L. Klanfer, _Metrische Charakterisierung der Kugel_ Ergebnisse math. Kolloquium Wien 4 (1933), 43-45.
  * [35] F. Luo, _3-dimensional Schläfli formula and its generalization_ Commun. Contemp. Math. 10, Suppl. 1 (2008), 835-842.
  * [36] K. Menger, _New foundations of Euclidean Geometry_ Am. J. Math. 53 (1931), 721-745.
  * [37] J. Milnor, _Collected papers. Vol. 1: Geometry._ Houston, TX: Publish or Perish. (1994).
  * [38] A. Neumaier, _Distance matrices, dimension, and conference graphs._ Indag. Math. 43 (1981), 385-391.
  * [39] R. Noda, T. Sakai, M. Morimoto, _Generalized Fermat’s problem._ Can. Math. Bull. 34, No. 1 (1991), 96-104.
  * [40] L. Schläfli, _Collected mathematical works. Vol. II. (Gesammelte mathematische Abhandlungen. Herausgegeben vom Steiner-Schläfli-Komitee der Schweizerischen Naturforschenden Gesellschaft. Bd. II.)_ (1953) Basel: Birkhäuser Verlag.
  * [41] I.J Schoenberg, _Remarks to Maurice Frechet’s article “Sur la definition axiomatique d’une classe d’espaces vectoriels distancies applicables vectoriellement sur l’espace de Hilbert”_ Ann. Math. (2) 36 (1935), 724-732.
  * [42] J. J. Seidel, _Quasiregular two-distance sets._ Nederl. Akad. Wet., Proc., Ser. A 72 (1969), 64-70.
  * [43] D. M. Y. Sommerville, _An Introduction to the Geometry of n Dimensions._ New York:Dover Publications, (1958).
  * [44] R. Sturm, Ueber den Punkt kleinster Entfernungsumme von gegebenen Punkten. J. Reine Angew. Math. XCVII (1884), 49-62.
  * [45] T. Tao, _The spherical Cayley-Menger determinant and the radius of the Earth_ , https://terrytao.wordpress.com, (2019).
  * [46] W.P. Thurston, _Three-dimensional geometry and topology. Vol. 1. Ed. by Silvio Levy._ Princeton Mathematical Series. 35 Princeton, NJ: Princeton University Press, (1997).
  * [47] V.A. Toponogov, _Differential geometry of curves and surfaces_ , Birkhäuser, 2005.
  * [48] J. V. Uspensky, _Theory of Equations_. New York McGraw-Hill, (1948).
  * [49] A. Zachos and G. Zouzoulas, The weighted Fermat-Torricelli problem for tetrahedra and an ”inverse” problem, _J. Math. Anal. Appl._ 353, (2009), no.1, 114-120.
  * [50] A. Zachos, _Location of the weighted Fermat-Torricelli point on the K -plane._ Analysis, München 33 (3) (2013), 243-249.
  * [51] A. N. Zachos, _Location of the weighted Fermat-Torricelli point on the K -plane. II._ Analysis, München 34 (1) (2014), 111-120.
  * [52] A. N. Zachos, An analytical solution of the weighted Fermat-Torricelli problem on a unit sphere. Rend. Circ. Mat. Palermo (2) 64 (2015), No. 3, 451-458.
  * [53] A. N. Zachos, _The weighted Fermat-Torricelli-Menger problem for a given sextuple of edge lengths determining tetrahedra._ J.Math. Chem. 54 (7), (2016), 1447-1460.
  * [54] A. Zachos, _A Generalized Fermat-Torricelli Tree in the Three Dimensional Euclidean Space_ J. Convex Anal., 24, No. 1 (2017) 185–198.
  * [55] A. N. Zachos, An evolutionary design of weighted minimum networks for four points in the three-dimensional Euclidean space, Analysis (Munchen) 41 (2) (2021), 79-112.