# Generalization of the Apollonius theorem for simplices and related problems

Michael N. Vrahatis  Department of Mathematics, University of Patras, GR-26110
Patras, Greece<EMAIL_ADDRESS>&<EMAIL_ADDRESS>
###### Abstract.

The Apollonius theorem gives the length of a median of a triangle in terms of
the lengths of its sides. The straightforward generalization of this theorem
obtained for _m_ -simplices in the _n_ -dimensional Euclidean space for _n_
greater than or equal to _m_ is given. Based on this, generalizations of
properties related to the medians of a triangle are presented. In addition,
applications of the generalized Apollonius’s theorem and the related to the
medians results, are given for obtaining: (a) the minimal spherical surface
that encloses a given simplex or a given bounded set, (b) the thickness of a
simplex that it provides a measure for the quality or how well shaped a
simplex is, and (c) the convergence and error estimates of the root-finding
bisection method applied on simplices.

###### Key words and phrases:

Euclidean geometries and generalizations, geometry and topology,
Generalization of Apollonius’ theorem, Generalization of Commandino’s theorem,
Generalization of Pythagoras’ theorem, Generalization of Carnot’s theorem,
simplex and set enclosing (covering), simplex thickness, simplex bisection,
minimum enclosing ball.

###### 2020 Mathematics Subject Classification:

Primary 51M05, 52A40. Secondary 52A35, 53A07, 51M20, 51M25, 52A37.

##  1\. Introduction

Apollonius111Apollonius of Perga (_c_. 240 – _c_. 190 B.C.), Greek
mathematician, known as the “Great Geometer”. gave a theorem known as the
_Apollonius theorem_ that relates the length of a median of a triangle in
terms of the lengths of its sides. Specifically, he gave the following well
known and widely used theorem:

######  Theorem 1.1 (Apollonius’ theorem).

In any triangle $\Delta ABC$ with vertices $A$, $B$, $C$ and lengths of its
sides $\ell(\,\,\overline{\\!\\!AB})$, $\ell(\,\,\overline{\\!\\!AC})$ and
$\ell(\,\,\overline{\\!\\!BC})$, if $\overline{\\!\\!AD}$ is a median with
length $\ell(\,\,\overline{\\!\\!AD})$, it holds that:

(1.1)
$2\bigl{[}\ell(\,\,\overline{\\!\\!AB})^{2}+\ell(\,\,\overline{\\!\\!AC})^{2}\bigr{]}=\ell(\,\,\overline{\\!\\!BC})^{2}+4\,\ell(\,\,\overline{\\!\\!AD})^{2}.$

The straightforward generalization of the concept of a triangle to the $n$
- dimensional Euclidean space $\mathbb{R}^{n}$ is the simplex, that is the
smallest possible $n$-dimensional polytope, _i.e._ the convex hull of a finite
number of points [36, p.8]. In general, in geometry, a simplex is considered
to be a generalization of the notion of a triangle to arbitrary dimensions.

Based on this notion we give the generalization of Apollonius’ theorem for
$m$-simplices in $\mathbb{R}^{n}$ ($n\geqslant m$) (see §2.2). Furthermore, we
give generalizations obtained for simplices of well known properties related
to the medians of triangles (see §2.3). Also, applications of the generalized
Apollonius’ theorem and the related results are given for obtaining: (a) the
minimal spherical surface that encloses a given simplex or a bounded subset of
$\mathbb{R}^{n}$ (see §3.1), (b) the thickness of a simplex that provides a
measure for the quality or how well shaped a simplex is (see §3.2) and (c) the
convergence and error estimates of the root-finding bisection method applied
on simplices (see §3.3). The provided proofs of the proposed results are
simple.

##  2\. Generalization of the Apollonius theorem for simplices

###  2.1. Notation, definitions, and preliminary concepts

######  Definition 2.1.

For any positive integer $n$, and for any set of points
$\\{{\upsilon}^{k}\\}_{k=0}^{n}$ in some linear space which are affinely
independent (_i.e._ the vectors
$\\{{\upsilon}^{k}-{\upsilon}^{0}\\}_{k=1}^{n}$ are linearly independent) the
convex hull ${\rm co}\\{{\upsilon}^{0},$ ${\upsilon}^{1},\ldots,$
${\upsilon}^{n}\\}=[{\upsilon}^{0},$ ${\upsilon}^{1},\ldots,$
${\upsilon}^{n}]$ is called the $n$-simplex with vertices ${\upsilon}^{0},$
${\upsilon}^{1},\ldots,{\upsilon}^{n}$. The convex hull of any nonempty subset
of the $n+1$ points that define an $n$-simplex is called a _face_ of the
simplex. Faces are simplices themselves and they are also called facets.
Specifically, for each subset of $m+1$ elements
$\\{{\omega}^{0},{\omega}^{1},\ldots,{\omega}^{m}\\}\subset\\{{\upsilon}^{0},{\upsilon}^{1},\ldots,{\upsilon}^{n}\\},$
the $m$-simplex $[{\omega}^{0},{\omega}^{1},\ldots,{\omega}^{m}]$ is called an
$m$-face of $[{\upsilon}^{0},$ ${\upsilon}^{1},\ldots,{\upsilon}^{n}].$ In
particular, 0-faces are vertices and 1-faces are edges. If all the edges have
the same length, the simplex is called _regular_ or _equilateral_ (_cf._
[10]).

######  Remark 2.2.

The order in which the points $\\{{\upsilon}^{k}\\}_{k=0}^{n}$ are written in
the lists determines an orientation. On the other hand, the convex hull is
independent of this order and thus the vertices can be permuted in a list.

######  Notation 2.3.

An $m$-simplex $\sigma^{m}$ in $\mathbb{R}^{n}$ ($n\geqslant m$) with set of
vertices $\\{{\upsilon}^{k}\\}_{k=0}^{m}$ is denoted by
$\sigma^{m}=[{\upsilon}^{0},{\upsilon}^{1},\ldots,{\upsilon}^{m}]$ and its
boundary by ${\vartheta}\kern 0.28436pt$${\sigma}^{m}$. Also, we denote the
$(m-1)$-simplex that determines the $i$-th $(m-1)$-face opposite to vertex
${\upsilon}^{i}$ of ${\sigma}^{m}$ by $\sigma^{m}_{\neg i}=[{\upsilon}^{0},$
${\upsilon}^{1},\ldots,$
${\upsilon}^{i-1},{\upsilon}^{i+1},\ldots,{\upsilon}^{m}]$. Similarly, we
denote the $(m-2)$-simplex that determines the $j$-th $(m-2)$-face opposite to
vertex ${\upsilon}^{j}$ of $\sigma^{m}_{\neg i}$, for $j\neq i$, by
$\sigma^{m}_{\neg ij}=[{\upsilon}^{0},$ ${\upsilon}^{1},\ldots,$
${\upsilon}^{i-1},{\upsilon}^{i+1},\ldots,$
${\upsilon}^{j-1},{\upsilon}^{j+1},\ldots,{\upsilon}^{m}]$.

######  Remark 2.4.

Obviously, it holds that ${\vartheta}\kern
0.28436pt$${\sigma}^{m}=\bigcup_{i=0}^{m}\sigma^{m}_{\neg i}$.

######  Remark 2.5.

The number of $k$-faces of an $m$-simplex $\sigma^{n}$ in $\mathbb{R}^{n}$
($n\geqslant m$) is equal to the binomial coefficient $\binom{m+1}{k+1}$
(_cf._ [21, p.120]). For example, the number of 0-faces (vertices) is
$\binom{m+1}{1}=m+1$, the number of 1-faces (edges) is
$\binom{m+1}{2}=m(m+1)/2$, while the number of $(m-1)$-faces is
$\binom{m+1}{m}=m+1$.

######  Definition 2.6.

The _diameter_ of an $m$-simplex $\sigma^{m}$ in $\mathbb{R}^{n}$ ($n\geqslant
m$) denoted by ${\rm diam}(\sigma^{m})$, is the length of the longest edge
(1-face) of $\sigma^{m}$ (_cf._ [3, p.607], [90, p.812]), where the Euclidean
norm, $\|\cdot\|$, is used here to measure distances. The length of the
shortest edge of $\sigma^{m}$ will be denoted by ${\rm shor}(\sigma^{m})$. The
_diameter_ of a bounded set $P$ in $\mathbb{R}^{n}$ is defined as ${\rm
diam}(P)=\sup_{x,y\in P}\|x-y\|$, while the _width_ or _breadth_ of $P$
denoted by ${\rm wid}(P)$, is the minimum distance of two parallel supporting
hyperplanes (tac-hyperplanes) of $P$.

######  Definition 2.7.

Let $\sigma^{m}=[{\upsilon}^{0},{\upsilon}^{1},\ldots,{\upsilon}^{m}]$ be an
$m$-simplex in $\mathbb{R}^{n}$ ($n\geqslant m$) then, the _barycenter_ or
_centroid_ of $\sigma^{m}$ and the _barycenter_ or _centroid_ of the $i$-th
$(m-1)$-face $\sigma^{m}_{\neg i}$, $i=0,1,\ldots,m$, of $\sigma^{m}$ are
respectively denoted by $\kappa^{m}$ and $\kappa_{i}^{m}$ and are given by

(2.1) $\kappa^{m}=\frac{1}{m+1}\sum_{j=0}^{m}{\upsilon}^{j}\,\,\text{ \ and \
}\,\,\kappa_{i}^{m}=\frac{1}{m}\sum_{\begin{subarray}{c}j=0\\\ j\neq
i\end{subarray}}^{m}{\upsilon}^{j},\,\,\,\,i=0,1,\ldots,m.$

######  Remark 2.8.

By convexity the barycenter of any $m$-simplex $\sigma^{m}$ in
$\mathbb{R}^{n}$ is a point in the relative interior of $\sigma^{m}$.

######  Definition 2.9.

Let $\sigma^{m}=[{\upsilon}^{0},{\upsilon}^{1},\ldots,{\upsilon}^{m}]$ be an
$m$-simplex in $\mathbb{R}^{n}$ ($n\geqslant m$) then, $\sigma^{m}$ is called
_orthocentric_ if its $m+1$ altitudes intersect in a common point $o^{m}$,
called its _orthocenter_.

######  Remark 2.10.

The $m+1$ altitudes of an $m$-simplex are not necessarily concurrent if
$m\geqslant 3$.

######  Definition 2.11.

Let $\sigma^{m}=[{\upsilon}^{0},{\upsilon}^{1},\ldots,{\upsilon}^{m}]$ be an
$m$-simplex in $\mathbb{R}^{n}$ ($n\geqslant m$) and let $\kappa_{i}^{m}$ be
the barycenter of the $i$-th $(m-1)$-face $\sigma^{m}_{\neg i}$, opposite to
vertex ${\upsilon}^{i}$ of $\sigma^{m}$ then, the 1-simplex
${\mu}_{i}^{m}=\bigl{[}{\upsilon}^{i},\,\kappa_{i}^{m}\bigr{]}$ for
$i=0,1,\ldots,m$ is called the _$i$ -th median_ of $\sigma^{m}$ that
corresponds to the vertex ${\upsilon}^{i}$.

######  Notation 2.12.

The $(n-1)$-dimensional spherical surface is denoted by ${S}^{n-1}$ for which
${S}^{n-1}=\vartheta{B}^{n}_{r,c}$ where ${B}^{n}_{r,c}\subset\mathbb{R}^{n}$
denotes the standard closed Euclidean ball of radius $r>0$ centered at a point
$c$ in $\mathbb{R}^{n}$, _i.e._
${B}^{n}_{r,c}=\\{x\in\mathbb{R}^{n}:\|x-c\|\leqslant r\\}$.

######  Definition 2.13.

Assume that $\sigma^{m}$ is an $m$-simplex in ${\mathbb{R}}^{n}$ $(n\geqslant
m)$ and let $\kappa^{m}$ be its barycenter. Then the smalest radius of the
spherical surfaces ${S}^{n-1}$ that enclose $\sigma^{m}$ is called
_circumradius_ and it is denoted by $\rho_{\rm cir}^{m}$ and the corresponding
center denoted by $c_{\rm cir}^{m}$ is called _circumcenter_. If $c_{\rm
cir}^{m}=\kappa^{m}$ then we call the corresponding circumradius as
_barycentric circumradius_ and we denote it as $\beta_{\rm cir}^{m}$.

######  Definition 2.14.

Let $\sigma^{m}$ be an $m$-simplex in ${\mathbb{R}}^{n}$ $(n\geqslant m)$ with
diameter ${\rm diam}(\sigma^{m})$ and barycenter $\kappa^{m}$. Then, the
largest radius of the spherical surfaces ${S}^{n-1}$ that are enclosed within
$\sigma^{m}$ is called _inradius_ of $\sigma^{m}$, it is denoted by $\rho_{\rm
inr}^{m}$ and the corresponding center denoted by $c_{\rm inr}^{m}$ is called
_incenter_. If $c_{\rm inr}^{m}=\kappa^{m}$ then the corresponding inradius
denoted by $\beta_{\rm inr}^{m}$ is called _barycentric inradius_ and it is
given by:

(2.2) $\beta_{\rm
inr}^{m}=\min_{x\in{\vartheta}{\sigma}^{m}}\|\kappa^{m}-x\|,$

where ${\vartheta}{\sigma}^{m}$ denotes the boundary of $\sigma^{m}$.
Furthermore, the following dimensionless quantity

(2.3) ${\theta}({\sigma}^{m})=\beta_{\rm inr}^{m}/{\rm diam}(\sigma^{m}),$

is called the _thickness_ of $\sigma^{m}$ (_cf._ [46, 55, 68, 90]).

Next, the straightforward generalization of the Apollonius theorem for
$m$-simplices in ${\mathbb{R}}^{n}$ $(n\geqslant m)$ follows.

###  2.2. Generalization of the Apollonius theorem

#####
Apollonius’ theorem (_cf._ Theorem 1.1) can be formulated for 2-simplices as
follows:

######  Theorem 2.15 (Apollonius’ theorem for 2-simplices).

Let $\sigma^{2}=[{\upsilon}^{0},{\upsilon}^{1},{\upsilon}^{2}]$ be a
$2$-simplex in $\mathbb{R}^{n}$ $(n\geqslant 2)$. If $\mu_{i}^{2}$ is the
$i$-th median of $\sigma^{2}$ that corresponds to the vertex ${\upsilon}^{i}$,
$i=0,1,2$, it holds that:

(2.4) $2\sum_{\begin{subarray}{c}j=0\\\ j\neq
i\end{subarray}}^{2}\|\upsilon^{i}-\upsilon^{j}\|^{2}=\sum_{\begin{subarray}{c}p=0\\\
p\neq i\end{subarray}}^{1}\sum_{\begin{subarray}{c}q=p+1\\\ q\neq
i\end{subarray}}^{2}\|\upsilon^{p}-\upsilon^{q}\|^{2}+2^{2}\,\|\mu_{i}^{2}\|^{2},\quad
i=0,1,2.$

######  Example 2.16.

By setting in Eq. (2.4) $i=0$ it follows that:

$2\bigl{(}\|{\upsilon}^{0}-{\upsilon}^{1}\|^{2}+\|{\upsilon}^{0}-{\upsilon}^{2}\|^{2}\bigr{)}=\|{\upsilon}^{1}-{\upsilon}^{2}\|^{2}+2^{2}\,\|\mu_{0}^{2}\|^{2},$

(_cf._ Theorem 1.1).

######  Theorem 2.17 (Generalization of the Apollonius theorem for simplices).

Assume that $\sigma^{m}=[{\upsilon}^{0},{\upsilon}^{1},\ldots,{\upsilon}^{m}]$
is an $m$-simplex in ${\mathbb{R}}^{n}$ $(n\geqslant m)$ with barycenter
$\kappa^{m}$. Assume further that $\kappa_{i}^{m}$ is the barycenter of the
$i$-th $(m-1)$-face $\sigma^{m}_{\neg i}$ of $\sigma^{m}$ and let
$\mu_{i}^{m}=[\kappa_{i},\,{\upsilon}^{i}]$ be the $i$-th median that
corresponds to the vertex ${\upsilon}^{i}$ for $i=0,1,\ldots,m$. Then it holds
that:

(2.5) $m\sum_{\begin{subarray}{c}j=0\\\ j\neq
i\end{subarray}}^{m}\|\upsilon^{i}-\upsilon^{j}\|^{2}=\sum_{\begin{subarray}{c}p=0\\\
p\neq i\end{subarray}}^{m-1}\sum_{\begin{subarray}{c}q=p+1\\\ q\neq
i\end{subarray}}^{m}\|\upsilon^{p}-\upsilon^{q}\|^{2}+m^{2}\|\mu_{i}^{m}\|^{2},\quad
i=0,1,\ldots,m.$

###### Proof.

Assume that
${\upsilon}^{i}=({\upsilon}^{i}_{1},{\upsilon}^{i}_{2},\ldots,{\upsilon}^{i}_{n})$
is a vertex of $\sigma^{m}$, then we have
$\|\mu_{i}^{m}\|^{2}=\|\kappa_{i}-{\upsilon}^{i}\|^{2}$. Equivalently, we can
obtain

(2.6) $\|\mu_{i}^{m}\|^{2}={\frac{1}{m^{2}}}\,\mathit{\Psi}_{i},$

where

$\mathit{\Psi}_{i}=\sum^{n}_{t=1}\,\Biggl{\\{}\,\sum_{\begin{subarray}{c}l=0\\\
l\neq
i\end{subarray}}^{m}{\upsilon}^{l}_{t}-m{\upsilon}^{i}_{t}\Biggr{\\}}^{2}.$

The term $\mathit{\Psi}_{i}$ can be written as follows:

$\displaystyle\mathit{\Psi}_{i}=\sum^{n}_{t=1}\Biggl{\\{}\biggl{[}\sum_{\begin{subarray}{c}l=0\\\
l\neq
i\end{subarray}}^{m}{\upsilon}^{l}_{t}\biggr{]}^{2}+m^{2}\bigl{[}{\upsilon}^{i}_{t}\bigr{]}^{2}-2m{\upsilon}^{i}_{t}\sum_{\begin{subarray}{c}l=0\\\
l\neq i\end{subarray}}^{m}{\upsilon}^{l}_{t}\Biggr{\\}}\Longleftrightarrow$
$\displaystyle\mathit{\Psi}_{i}=\sum^{n}_{t=1}\Biggl{\\{}\sum_{\begin{subarray}{c}l=0\\\
l\neq
i\end{subarray}}^{m}\bigl{[}{\upsilon}^{l}_{t}\bigr{]}^{2}+2\sum_{\begin{subarray}{c}p=0\\\
p\neq i\end{subarray}}^{m-1}\sum_{\begin{subarray}{c}q=p+1\\\ q\neq
i\end{subarray}}^{m}{\upsilon}^{p}_{t}{\upsilon}^{q}_{t}+m^{2}\bigl{[}{\upsilon}^{i}_{t}\bigr{]}^{2}-2m{\upsilon}^{i}_{t}\sum_{\begin{subarray}{c}l=0\\\
l\neq i\end{subarray}}^{m}{\upsilon}^{l}_{t}\Biggr{\\}},$

from which we obtain

$\displaystyle\mathit{\Psi}_{i}=\sum^{n}_{t=1}\Biggl{\\{}m\sum_{\begin{subarray}{c}l=0\\\
l\neq
i\end{subarray}}^{m}\bigl{[}{\upsilon}^{l}_{t}\bigr{]}^{2}+m^{2}\bigl{[}{\upsilon}^{i}_{t}\bigr{]}^{2}-2m{\upsilon}^{i}_{t}\sum_{\begin{subarray}{c}l=0\\\
l\neq
i\end{subarray}}^{m}{\upsilon}^{l}_{t}-(m-1)\sum_{\begin{subarray}{c}l=0\\\
l\neq i\end{subarray}}^{m}\bigl{[}{\upsilon}^{l}_{t}\bigr{]}^{2}$
$\displaystyle\kern 52.63777pt+2\sum_{\begin{subarray}{c}p=0\\\ p\neq
i\end{subarray}}^{m-1}\sum_{\begin{subarray}{c}q=p+1\\\ q\neq
i\end{subarray}}^{m}{\upsilon}^{p}_{t}{\upsilon}^{q}_{t}\Biggr{\\}}\Longleftrightarrow$
$\displaystyle\mathit{\Psi}_{i}=m\sum^{n}_{t=1}\Biggl{\\{}\sum_{\begin{subarray}{c}l=0\\\
l\neq
i\end{subarray}}^{m}\bigl{[}{\upsilon}^{l}_{t}\bigr{]}^{2}+m\bigl{[}{\upsilon}^{i}_{t}\bigr{]}^{2}-2{\upsilon}^{i}_{t}\sum_{\begin{subarray}{c}l=0\\\
l\neq i\end{subarray}}^{m}{\upsilon}^{l}_{t}\Biggr{\\}}$ $\displaystyle\kern
52.63777pt-\sum^{n}_{t=1}\Biggl{\\{}(m-1)\sum_{\begin{subarray}{c}l=0\\\ l\neq
i\end{subarray}}^{m}\bigl{[}{\upsilon}^{l}_{t}\bigr{]}^{2}-2\sum_{\begin{subarray}{c}p=0\\\
p\neq i\end{subarray}}^{m-1}\sum_{\begin{subarray}{c}q=p+1\\\ q\neq
i\end{subarray}}^{m}{\upsilon}^{p}_{t}{\upsilon}^{q}_{t}\Biggr{\\}},$

or equivalently, after some algebraic manipulations, we obtain

(2.7) $\mathit{\Psi}_{i}=m\sum_{\begin{subarray}{c}j=0\\\ j\neq
i\end{subarray}}^{m}\|\upsilon^{i}-\upsilon^{j}\|^{2}-\sum_{\begin{subarray}{c}p=0\\\
p\neq i\end{subarray}}^{m-1}\sum_{\begin{subarray}{c}q=p+1\\\ q\neq
i\end{subarray}}^{m}\|\upsilon^{p}-\upsilon^{q}\|^{2}.$

Companying Eq. (2.6) and Eq. (2.7) we have

(2.8)
$\|\mu_{i}^{m}\|^{2}=\frac{1}{m^{2}}\Biggl{(}m\sum_{\begin{subarray}{c}j=0\\\
j\neq
i\end{subarray}}^{m}\|\upsilon^{i}-\upsilon^{j}\|^{2}-\sum_{\begin{subarray}{c}p=0\\\
p\neq i\end{subarray}}^{m-1}\sum_{\begin{subarray}{c}q=p+1\\\ q\neq
i\end{subarray}}^{m}\|\upsilon^{p}-\upsilon^{q}\|^{2}\Biggr{)}.$

Thus the theorem is proved. ∎

###  2.3. Generalizations of median properties

######  Remark 2.18.

The generalization of the Apollonius Theorem 2.17 provides the following
immediate result which gives an expression for the medians length of a simplex
that can be calculated in terms of its edge lengths.

######  Corollary 2.19 (Medians length).

Assume that $\sigma^{m}=[{\upsilon}^{0},{\upsilon}^{1},\ldots,{\upsilon}^{m}]$
is an $m$-simplex in ${\mathbb{R}}^{n}$ $(n\geqslant m)$. Then the lengths of
the $m+1$ medians $\mu_{i}^{m}$ for $i=0,1,\ldots,m$ of $\sigma^{m}$ are given
as follows:

(2.9) $\|\mu_{i}^{m}\|=\frac{1}{m}\Biggl{(}m\sum_{\begin{subarray}{c}j=0\\\
j\neq
i\end{subarray}}^{m}\|\upsilon^{i}-\upsilon^{j}\|^{2}-\sum_{\begin{subarray}{c}p=0\\\
p\neq i\end{subarray}}^{m-1}\sum_{\begin{subarray}{c}q=p+1\\\ q\neq
i\end{subarray}}^{m}\|\upsilon^{p}-\upsilon^{q}\|^{2}\Biggr{)}^{1/2}.$

###### Proof.

Since the right hand side of Eq. (2.8) is positive, the proof follows from Eq.
(2.8). ∎

######  Remark 2.20.

The length of the $i$-th median $\mu_{i}^{m}$ for $i=0,1,\ldots,m$ of
$\sigma^{m}$ that corresponds to the vertex ${\upsilon}^{i}$ is given in terms
of the lengths of (a) the $m$ edges of $\sigma^{m}$ that concur in the vertex
${\upsilon}^{i}$ and (b) the $\binom{m}{2}$ edges of the $i$-th $(m-1)$-face
$\sigma^{m}_{\neg i}$ of $\sigma^{m}$.

######  Remark 2.21.

It is well known the property of a triangle that its three medians concur in
the centroid that divides each of them in the ratio $1:2$, where the longer
segment being on the side of the vertex. In 1565 Commandino222 Federico
Commandino (1509 – 1575), Italian mathematician. in his work _“De centro
gravitates solidorum”_ (The center of gravity of solids), extended the above
property of triangles for the case of the medians of a tetrahedron (_i.e._ the
line segments that join each vertex of the tetrahedron to the barycenter of
the opposite face). The corresponding result is known as _Commandino’s
theorem_ [4, p.97].

######  Theorem 2.22 (Commandino’s theorem (1565)).

The four medians of a tetrahedron concur in a point that divides each of them
in the ratio $1:3$, where the longer segment being on the side of the vertex
of the tetrahedron.

Next, the generalization of Commandino’s theorem for $m$-simplices in
${\mathbb{R}}^{n}$ $(n\geqslant m)$ follows.

######  Theorem 2.23 (Generalization of Commandino’s theorem for simplices).

Assume that the hypotheses of Theorem 2.17 are fulfilled, then the $m+1$
medians of $\sigma^{m}$ concur in the barycenter $\kappa^{m}$ of $\sigma^{m}$
that divides each of them in the ratio $1:m$, where the longer segment being
on the side of the vertex of $\sigma^{m}$.

###### Proof.

Let
${\upsilon}^{i}=({\upsilon}^{i}_{1},{\upsilon}^{i}_{2},\ldots,{\upsilon}^{i}_{n})$
be a vertex of $\sigma^{m}$, then

$\displaystyle\kern-56.9055pt\kappa^{m}-{\upsilon}^{i}=\Bigl{\\{}$
$\displaystyle\Bigl{[}\frac{1}{m+1}\,\Bigl{(}{\upsilon}^{0}_{1}+{\upsilon}^{1}_{1}+\cdots+{\upsilon}^{m}_{1}\Bigr{)}-{\upsilon}^{i}_{1}\Bigr{]},\ldots,$
$\displaystyle\Bigl{[}\frac{1}{m+1}\,\Bigl{(}{\upsilon}^{0}_{n}+{\upsilon}^{1}_{n}+\cdots+{\upsilon}^{m}_{n}\Bigr{)}-{\upsilon}^{i}_{n}\Bigr{]}\Bigr{\\}}\Longleftrightarrow$
(2.10) $\displaystyle\kappa^{m}-{\upsilon}^{i}=\frac{1}{m+1}\,\Bigl{\\{}$
$\displaystyle\Bigl{(}{\upsilon}^{0}_{1}+{\upsilon}^{1}_{1}+\cdots+{\upsilon}^{i-1}_{1}-m{\upsilon}^{i}_{1}+{\upsilon}^{i+1}_{1}+\cdots+{\upsilon}^{m}_{1}\Bigr{)},\ldots,$
$\displaystyle\,\Bigl{(}{\upsilon}^{0}_{n}+{\upsilon}^{1}_{n}+\cdots+{\upsilon}^{i-1}_{n}-m{\upsilon}^{i}_{n}+{\upsilon}^{i+1}_{n}+\cdots+{\upsilon}^{m}_{n}\Bigr{)}\Bigr{\\}}.$

Similarly for the barycenter $\kappa^{m}_{i}$ of the oppositive to the vertex
${\upsilon}^{i}$ $(m-1)$-face $\sigma^{m}_{\neg i}$ of $\sigma^{m}$ we have

$\displaystyle\kern 14.22636pt\kappa^{m}_{i}-{\upsilon}^{i}=\Bigl{\\{}$
$\displaystyle\Bigl{[}\frac{1}{m}\,\Bigl{(}{\upsilon}^{0}_{1}+{\upsilon}^{1}_{1}+\cdots+{\upsilon}^{i-1}_{1}+{\upsilon}^{i+1}_{1}+\cdots+{\upsilon}^{m}_{1}\Bigr{)}-{\upsilon}^{i}_{1}\Bigr{]},\ldots,$
$\displaystyle\Bigl{[}\frac{1}{m}\,\Bigl{(}{\upsilon}^{0}_{n}+{\upsilon}^{1}_{n}+\cdots+{\upsilon}^{i-1}_{n}+{\upsilon}^{i+1}_{n}+\cdots+{\upsilon}^{m}_{n}\Bigr{)}-{\upsilon}^{i}_{n}\Bigr{]}\Bigr{\\}}\Longleftrightarrow$
(2.11) $\displaystyle\kappa^{m}_{i}-{\upsilon}^{i}=\frac{1}{m}\,\Bigl{\\{}$
$\displaystyle\Bigl{(}{\upsilon}^{0}_{1}+{\upsilon}^{1}_{1}+\cdots+{\upsilon}^{i-1}_{1}-m{\upsilon}^{i}_{1}+{\upsilon}^{i+1}_{1}+\cdots+{\upsilon}^{m}_{1}\Bigr{)},\ldots,$
$\displaystyle\,\Bigl{(}{\upsilon}^{0}_{n}+{\upsilon}^{1}_{n}+\cdots+{\upsilon}^{i-1}_{n}-m{\upsilon}^{i}_{n}+{\upsilon}^{i+1}_{n}+\cdots+{\upsilon}^{m}_{n}\Bigr{)}\Bigr{\\}}.$

Companying Eq. (2.10) and Eq. (2.11) we obtain

(2.12)
$(m+1)(\kappa^{m}-{\upsilon}^{i})=m(\kappa^{m}_{i}-{\upsilon}^{i}),\quad
i=0,1,\ldots,m.$

Thus, the points ${\upsilon}^{i}$, $\kappa^{m}$ and $\kappa^{m}_{i}$ are
collinear points and the $m+1$ medians ${\mu}_{i}^{m}$, $i=0,1,\ldots,m$ of
$\sigma^{m}$ concur in the barycenter $\kappa^{m}$ of $\sigma^{m}$. Next,
since
$\kappa^{m}_{i}-{\upsilon}^{i}=(\kappa^{m}_{i}-\kappa^{m})+(\kappa^{m}-{\upsilon}^{i})$,
using Eq. (2.12) we obtain

(2.13) $\|\kappa^{m}-{\upsilon}^{i}\|=m\|\kappa^{m}_{i}-\kappa^{m}\|,\quad
i=0,1,\ldots,m.$

Therefore, the barycenter $\kappa^{m}$ of $\sigma^{m}$ divides each median
${\mu}_{i}^{m}$, $i=0,1,\ldots,m$ in the ratio $1:m$, where the longer segment
being on the side of the vertex of $\sigma^{m}$. Thus the theorem is proved. ∎

The Apollonius Theorem 1.1 gives the following immediate interesting results
(_cf._ [41, p.68]).

######  Theorem 2.24 (Median properties for triangles).

In any triangle $\Delta ABC$ with vertices $A$, $B$, $C$ and lengths of its
sides $\ell(\,\,\overline{\\!\\!AB})$, $\ell(\,\,\overline{\\!\\!AC})$ and
$\ell(\,\,\overline{\\!\\!BC})$, if $\overline{\\!\\!AD}$,
$\overline{\\!\\!BE}$ and $\overline{\\!\\!CF}$ are its medians with lengths,
respectively, $\ell(\,\,\overline{\\!\\!AD})$, $\ell(\,\,\overline{\\!\\!BE})$
and $\ell(\,\,\overline{\\!\\!CF})$ and if $M$ is the median point of
intersection of the medians, then the following hold:

(2.14)
$\displaystyle\ell(\,\,\overline{\\!\\!AD})^{2}+\ell(\,\,\overline{\\!\\!BE})^{2}+\ell(\,\,\overline{\\!\\!CF})^{2}$
$\displaystyle=\frac{3}{4}\bigl{[}\ell(\,\,\overline{\\!\\!AB})^{2}+\ell(\,\,\overline{\\!\\!AC})^{2}+\ell(\,\,\overline{\\!\\!BC})^{2}\bigr{]},$
(2.15)
$\displaystyle\ell(\,\,\overline{\\!\\!MA})^{2}+\ell(\,\,\overline{\\!\\!MB})^{2}+\ell(\,\,\overline{\\!\\!MC})^{2}$
$\displaystyle=\frac{1}{3}\bigl{[}\ell(\,\,\overline{\\!\\!AB})^{2}+\ell(\,\,\overline{\\!\\!AC})^{2}+\ell(\,\,\overline{\\!\\!BC})^{2}\bigr{]}.$

Straightforward generalizations of the above results are given as follows.

######  Theorem 2.25 (Median properties for simplices).

Let $\sigma^{m}=[{\upsilon}^{0},{\upsilon}^{1},\ldots,{\upsilon}^{m}]$ be an
$m$-simplex in ${\mathbb{R}}^{n}$ $(n\geqslant m)$ with barycenter
$\kappa^{m}$. Assume that $\kappa_{i}^{m}$ is the barycenter of the $i$-th
$(m-1)$-face $\sigma^{m}_{\neg i}$ of $\sigma^{m}$ and let
$\mu_{i}^{m}=[{\upsilon}^{i},\,\kappa_{i}^{m}]$ be the $i$-th median that
corresponds to the vertex ${\upsilon}^{i}$ for $i=0,1,\ldots,m$. Then the
following hold:

(2.16) $\displaystyle\sum_{i=0}^{m}\|\mu_{i}^{m}\|^{2}$
$\displaystyle=\frac{m+1}{m^{2}}\sum_{p=0}^{m-1}\sum_{q=p+1}^{m}\|{\upsilon}^{p}-{\upsilon}^{q}\|^{2},$
(2.17) $\displaystyle\sum_{i=0}^{m}\|\kappa^{m}-{\upsilon}^{i}\|^{2}$
$\displaystyle=\frac{1}{m+1}\sum_{p=0}^{m-1}\sum_{q=p+1}^{m}\|{\upsilon}^{p}-{\upsilon}^{q}\|^{2}.$

###### Proof.

The proof of Eq. (2.16) can be obtained using Eq. (2.8) for $i=0,1,\ldots,m$
and by adding the corresponding expressions to obtain the term
$\sum_{i=0}^{m}\|\mu_{i}^{m}\|^{2}$. Since an edge
$[{\upsilon}^{i},{\upsilon}^{j}]$ of $\sigma^{m}$ corresponds to the two
vertices ${\upsilon}^{i}$ and ${\upsilon}^{j}$ of $\sigma^{m}$ then for the
first term of the right hand side of Eq. (2.8) we have

(2.18) $\sum_{i=0}^{m}\sum_{\begin{subarray}{c}j=0\\\ j\neq
i\end{subarray}}^{m}\|\upsilon^{i}-\upsilon^{j}\|^{2}=2\sum_{p=0}^{m-1}\sum_{q=p+1}^{m}\|{\upsilon}^{p}-{\upsilon}^{q}\|^{2},$

since each edge is added twice which results to twice the sum of the squares
of the lengths of all the edges of $\sigma^{m}$.

Also, since an edge $[{\upsilon}^{i},{\upsilon}^{j}]$ of $\sigma^{m}$ can
constitute an edge of the $k$-th $(m-1)$-face $\sigma^{m}_{\neg k}$ of
$\sigma^{m}$ for the $m-1$ vertices ${\upsilon}^{k}$ of $\sigma^{m}$ where $k$
is different from $i$ and $j$, then for the second term of the right hand side
of Eq. (2.8) we have

(2.19) $-\sum_{i=0}^{m}\sum_{\begin{subarray}{c}p=0\\\ p\neq
i\end{subarray}}^{m-1}\sum_{\begin{subarray}{c}q=p+1\\\ q\neq
i\end{subarray}}^{m}\|\upsilon^{p}-\upsilon^{q}\|^{2}=-(m-1)\sum_{p=0}^{m-1}\sum_{q=p+1}^{m}\|{\upsilon}^{p}-{\upsilon}^{q}\|^{2},$

since each edge is added $m-1$ times which results to $m-1$ times the sum of
the squares of the lengths of all the edges of $\sigma^{m}$.

Companying Eq. (2.8), Eq. (2.18) and Eq. (2.19) we obtain

$\sum_{i=0}^{m}\|\mu_{i}^{m}\|^{2}=\frac{1}{m^{2}}\Biggl{(}2m\sum_{p=0}^{m-1}\sum_{q=p+1}^{m}\|{\upsilon}^{p}-{\upsilon}^{q}\|^{2}-(m-1)\sum_{p=0}^{m-1}\sum_{q=p+1}^{m}\|{\upsilon}^{p}-{\upsilon}^{q}\|^{2}\Biggr{)},$

which proves Eq. (2.16). Next, using Eq. (2.12), we have

(2.20)
$\|\mu_{i}^{m}\|^{2}=\|\kappa^{m}_{i}-{\upsilon}^{i}\|^{2}=\frac{(m+1)^{2}}{m^{2}}\|\kappa^{m}-{\upsilon}^{i}\|^{2},\quad\mbox{
for }\quad i=0,1,\ldots,m.$

Companying the above relations and Eq. (2.16) we obtain Eq. (2.17). Thus the
theorem is proved. ∎

######  Remark 2.26.

We note in passing that, Eq. (2.17) is related to the well known _Fermat’s
problem_ [22, 35, 41]. Specifically, in 1643 Fermat333 Pierre de Fermat (1607
– 1665), French mathematician., motivated by a letter of Descartes444 René
Descartes (1596 – 1650), French philosopher and mathematician., proposed the
following problem: “ _Find a point $F$ for which the sum of the Euclidean
distances from $F$ to the vertices $A$, $B$ and $C$ of a given triangle
$\Delta ABC$ is minimized._” This problem was first raised by Fermat in a
private letter to Torricelli555 Evangelista Torricelli (1608 – 1647), Italian
physicist and mathematician. who solved it. Torricelli’s solution was
published in 1659 by his student Viviani666 Vincenzo Viviani (1622 – 1703),
Italian mathematician.. Therefor, _Fermat’s point_ $F$ of a triangle is also
called _Torricelli point_ or _Fermat-Torricelli point_. Notice that, the
Fermat point $F$ of an equilateral (regular) triangle coincides with the
barycenter (centroid) $\kappa^{2}$ of the triangle (_cf._ Eq. (2.17)). Also,
it can be easily proved that in the case of regular simplices $\sigma^{m}$ in
${\mathbb{R}}^{n}$ with diameter ${\rm diam}(\sigma^{m})$ it holds that
$\sum_{i=0}^{m}\|\kappa^{m}-{\upsilon}^{i}\|=\sqrt{m(m+1)/2}\,\,{\rm
diam}(\sigma^{m})$ (_cf_. Corollary 3.8). It is worth noting that, in 1977
Alexander [2] has shown that only the regular simplex has minimal diameter
among all simplices inscribed in a sphere of ${\mathbb{R}}^{n}$ and containing
the center of the sphere.

######  Remark 2.27.

It is worth mentioning that, the regular simplex besides its symmetry exhibits
interesting and useful properties including, among others, the maximizing
properties of the regular simplex that have been given by Tanner in 1974 [76].
Specifically, Tanner has shown that the regular simplex maximizes the sum of
the squared contents of all $m$-faces, for all $m$, when the sum of squared
edge (line) lengths is fixed. Thus, the regular simplex has the largest total
length of all edges (joining lines), total area of all 2-faces (triangles),
total volume of all 3-faces (tetrahedra), _etc._ , for a fixed sum of squared
edge lengths [76]. In addition, in 1977 Alexander [2] proved the conjecture of
Sallee777 George Thomas Sallee (1940 – 2019), American mathematician. which
states that: _“Only the regular simplex has maximal width among all simplices
inscribed in a sphere of ${\mathbb{R}}^{n}$.”_ Note that, the width of the
regular simplex in ${\mathbb{R}}^{n}$ with diameter 1 is approximately
$\sqrt{2/n}$ and specifically $\sqrt{2/(n+1)}$ if $n$ is odd and
$\sqrt{2(n+1)}/\sqrt{n(n+2)}$ if $n$ is even (_cf_. [2], [33], [37]), for a
short proof, see Theorem 3.42. In addition, regular simplices are orthocentric
(_cf_. Theorem 2.28).

######  Theorem 2.28 (Medians of regular simplices).

Let $\sigma^{m}=[{\upsilon}^{0},{\upsilon}^{1},\ldots,{\upsilon}^{m}]$ be a
regular $m$-simplex in ${\mathbb{R}}^{n}$ $(n\geqslant m)$ with edge length
${\rm diam}(\sigma^{m})$ and barycenter $\kappa^{m}$. Let $\kappa_{i}^{m}$ be
the barycenter of the $i$-th $(m-1)$-face $\sigma^{m}_{\neg i}$ of
$\sigma^{m}$ and let $\mu_{i}^{m}=[{\upsilon}^{i},\,\kappa_{i}^{m}]$ be the
$i$-th median that corresponds to the vertex ${\upsilon}^{i}$ for
$i=0,1,\ldots,m$. Then, a) the $i$-th median of $\sigma^{m}$ coincides with
its corresponding $i$-th altitude, b) $\sigma^{m}$ is orthocentric and c) the
orthocenter $o^{m}$ of $\sigma^{m}$ coincides with its barycenter
$\kappa^{m}$.

###### Proof.

By Eq. (2.8) for the $i$-th median we obtain

$\|{\upsilon}^{i}-\kappa_{i}^{m}\|^{2}=\frac{1}{m^{2}}\left(m^{2}-\binom{m}{2}\right){\rm
diam}(\sigma^{m})^{2},\quad i=0,1,\ldots,m,$

or equivalently we have

(2.21) $\|{\upsilon}^{i}-\kappa_{i}^{m}\|^{2}=\frac{m+1}{2m}\,{\rm
diam}(\sigma^{m})^{2},\quad i=0,1,\ldots,m.$

Let $\mu_{ij}^{m}=[{\upsilon}^{j},\,\kappa_{ij}^{m}]$ be the $j$-th median,
$j=0,1,\ldots,m$, $j\neq i$, that corresponds to the vertex ${\upsilon}^{j}$
of the $i$-th $(m-1)$-face $\sigma^{m}_{\neg i}$. Then, similarly to the above
we have

(2.22) $\|{\upsilon}^{j}-\kappa_{ij}^{m}\|^{2}=\frac{m}{2(m-1)}\,{\rm
diam}(\sigma^{m})^{2}.$

Using Eq. (2.12) we obtain

(2.23)
$\|{\upsilon}^{j}-\kappa_{i}^{m}\|^{2}=\frac{(m-1)^{2}}{m^{2}}\|{\upsilon}^{j}-\kappa_{ij}^{m}\|^{2}.$

Companying Eqs. (2.22) and (2.23) we have

(2.24) $\|{\upsilon}^{j}-\kappa_{i}^{m}\|^{2}=\frac{m-1}{2m}\,{\rm
diam}(\sigma^{m})^{2},\quad j=0,1,\ldots,m,\kern 8.5359ptj\neq i.$

Companying Eqs. (2.21) and (2.24) we obtain

$\|{\upsilon}^{i}-\kappa_{i}^{m}\|^{2}+\|{\upsilon}^{j}-\kappa_{i}^{m}\|^{2}={\rm
diam}(\sigma^{m})^{2}.$

Since $\|{\upsilon}^{i}-{\upsilon}^{j}\|^{2}={\rm diam}(\sigma^{m})^{2}$ we
have

(2.25)
$\|{\upsilon}^{i}-\kappa_{i}^{m}\|^{2}+\|{\upsilon}^{j}-\kappa_{i}^{m}\|^{2}=\|{\upsilon}^{i}-{\upsilon}^{j}\|^{2}.$

Thus, by the Pythagorean theorem for right triangles for the 2-simplex
$[{\upsilon}^{i},\kappa_{i}^{m},{\upsilon}^{j}]$ we obtain that the $i$-th
median $\mu_{i}^{m}=[{\upsilon}^{i},\kappa_{i}^{m}]$ of $\sigma^{m}$ is also
its $i$-th altitude. Also, due to Theorem 2.23, $\sigma^{m}$ is orthocentric
and its orthocenter $o^{m}$ coincides with its barycenter $\kappa^{m}$. Thus
the theorem is proved. ∎

######  Remark 2.29.

Allow us to point out that, the Pythagorean theorem for right triangles due to
Pythagoras888 Pythagoras of Samos (_c_. 570 – _c_. 495 B.C.), Greek
mathematician and philosopher. states that: _“The sum of the squares on the
legs of a right triangle is equal to the square on the hypotenuse (the side
opposite the right angle).”_ This theorem is included by Euclid999 Euclid
(_c_. 323 – _c_. 285 B.C.), Greek mathematician, considered as the “Father of
Geometry”. in his seminal work _“Elements”_ and specifically in Book I,
Proposition 47 (_cf._ [29]).

######  Remark 2.30.

Eq. (2.25) could be named, the _generalized Pythagorean equation for regular
simplices_.

Based on the above, in conclusion, we point out the following theorem.

######  Theorem 2.31 (Generalization of Pythagoras’ theorem for regular
simplices).

Let $\sigma^{m}=[{\upsilon}^{0},{\upsilon}^{1},\ldots,{\upsilon}^{m}]$ be a
regular $m$-simplex in ${\mathbb{R}}^{n}$ $(n\geqslant m)$ and let
$\kappa_{i}^{m}$ for $i=0,1,\ldots,m$ be the barycenter of the $i$-th
$(m-1)$-face $\sigma^{m}_{\neg i}$ opposite to vertex ${\upsilon}^{i}$ of
$\sigma^{m}$. Then it holds that

(2.26)
$\|{\upsilon}^{i}-\kappa_{i}^{m}\|^{2}+\|{\upsilon}^{j}-\kappa_{i}^{m}\|^{2}=\|{\upsilon}^{i}-{\upsilon}^{j}\|^{2},\quad
i,j=0,1,\ldots,m,\kern 8.5359ptj\neq i.$

###### Proof.

The proof follows from Eq. (2.25). ∎

######  Remark 2.32.

In 1803 Carnot101010 Lazare Nicolas Marguerite Carnot (1753 – 1823), French
mathematician, physicist and politician. [19] gave a theorem known as
_Carnot’s theorem_ which states that: _“The sum of the distances from the
circumcenter to the three sides of a triangle is equal to the radius
(circumradius) of the circumscribed circle plus the radius (inradius) of the
inscribed circle”_ [62]. The most known application of Carnot’s theorem is in
the proof of the _Japanese theorem for a cyclic polygon_ (_i.e._ polygon with
vertices upon which a circle can be circumscribed) which states that: _“In any
triangulation of a cyclic polygon, the sum of inradii of triangles is
constant_ ”. We note in passing that, according to Hayashi111111 Tsuruichi
Hayashi (1873 – 1935), Japanese mathematician and historian.: “It was the
ancient custom of Japanese mathematicians to inscribe their discoveries on
tablets which were hung in the temples, to the glory of the gods and the honor
of the authors”. The Japanese theorem for cyclic polygons has been exhibited
in 1800 [41, p.193] and thus it is also known as _Japanese temple theorem_.

Based on the above we give the following generalization of Carnot’s theorem
for regular simplices.

######  Theorem 2.33 (Generalization of Carnot’s theorem for regular
simplices).

Let $\sigma^{m}=[{\upsilon}^{0},{\upsilon}^{1},\ldots,{\upsilon}^{m}]$ be a
regular $m$-simplex in ${\mathbb{R}}^{n}$ $(n\geqslant m)$ with barycenter
$\kappa^{m}$ and let $\kappa_{i}^{m}$ for $i=0,1,\ldots,m$ be the barycenter
of the $i$-th $(m-1)$-face $\sigma^{m}_{\neg i}$ opposite to vertex
${\upsilon}^{i}$ of $\sigma^{m}$. Then
(a) the distance from the barycenter to the $j$-th $(m-1)$-face
$\sigma^{m}_{\neg j}$ is given by

(2.27) $\min_{x\in\sigma^{m}_{\neg
j}}\|\kappa^{m}-x\|=\|\kappa^{m}-\kappa_{j}^{m}\|,\quad\forall\,j=0,1,\ldots,m,$

(b) the length of the barycentric circumradius $\beta_{\rm cir}^{m}$ of
$\sigma^{m}$ is given by

(2.28) $\beta_{\rm
cir}^{m}=\|{\upsilon}^{j}-\kappa^{m}\|,\quad\forall\,j=0,1,\ldots,m,$

(c) the length of the barycentric inradius $\beta_{\rm inr}^{m}$ of
$\sigma^{m}$ is given by

(2.29) $\beta_{\rm
inr}^{m}=\|\kappa^{m}-\kappa_{j}^{m}\|,\quad\forall\,j=0,1,\ldots,m,$

and (d) it holds that

(2.30) $\sum_{i=0}^{m}\|\kappa^{m}-\kappa_{i}^{m}\|=\beta_{\rm
cir}^{m}+\beta_{\rm inr}^{m}.$

###### Proof.

It is evident that in a regular simplex all medians are of equal length (_cf._
Eq. (2.21)) and in general by Theorem 2.23 concur in the barycenter that
divides each of them in the ratio $1:m$. Thus the proof can be obtained by
Theorem 2.28. ∎

######  Remark 2.34.

To clarify the notations in Theorem 2.33, we note that in the case of regular
$m$-simplices the barycenter $\kappa^{m}$ is the same with the circumcenter
$c_{\rm cir}^{m}$ and the incenter $c_{\rm inr}^{m}$. Also, the barycentric
circumradius $\beta_{\rm cir}^{m}$ is the same with the circumradius
$\rho_{\rm cir}^{m}$ while the barycentric inradius $\beta_{\rm inr}^{m}$ is
the same with the inradius $\rho_{\rm inr}^{m}$.

##  3\. Applications of the generalized Apollonius theorem

###  3.1. Simplex and set enclosing

######  Remark 3.1.

The generalization of the Apollonius theorem for simplices can be applied for
obtaining an estimation of the radius of a spherical surface that encloses a
given simplex or a bounded subset of $\mathbb{R}^{n}$.

######  Lemma 3.2.

Assume that $\sigma^{m}=[{\upsilon}^{0},{\upsilon}^{1},\ldots,{\upsilon}^{m}]$
is an $m$-simplex in ${\mathbb{R}}^{n}$ $(n\geqslant m)$ and let $\kappa^{m}$
be its barycenter. Then for each $i=0,1,\ldots,m$ it holds that:

(3.1)
$\|\kappa^{m}-{\upsilon}^{i}\|=\frac{1}{m+1}\Biggl{(}m\sum_{\begin{subarray}{c}j=0\\\
j\neq
i\end{subarray}}^{m}\|\upsilon^{i}-\upsilon^{j}\|^{2}-\sum_{\begin{subarray}{c}p=0\\\
p\neq i\end{subarray}}^{m-1}\sum_{\begin{subarray}{c}q=p+1\\\ q\neq
i\end{subarray}}^{m}\|\upsilon^{p}-\upsilon^{q}\|^{2}\Biggr{)}^{1/2}.$

###### Proof.

The proof follows by companying Eq. (2.8) and Eq. (2.12). ∎

######  Theorem 3.3 (Simplex barycentric enclosing).

Let $\sigma^{m}=[{\upsilon}^{0},{\upsilon}^{1},\ldots,{\upsilon}^{m}]$ be an
$m$-simplex in ${\mathbb{R}}^{n}$ $(n\geqslant m)$ and let $\kappa^{m}$ be its
barycenter. Then the barycentric circumradius $\beta_{\rm cir}^{m}$ is unique,
it is given as follows:

(3.2) $\beta_{\rm cir}^{m}=\frac{1}{m+1}\max_{0\leqslant i\leqslant
m}\Biggl{\\{}m\sum_{\begin{subarray}{c}j=0\\\ j\neq
i\end{subarray}}^{m}\|\upsilon^{i}-\upsilon^{j}\|^{2}-\sum_{\begin{subarray}{c}p=0\\\
p\neq i\end{subarray}}^{m-1}\sum_{\begin{subarray}{c}q=p+1\\\ q\neq
i\end{subarray}}^{m}\|\upsilon^{p}-\upsilon^{q}\|^{2}\Biggr{\\}}^{1/2},$

and the spherical surface with center at $\kappa^{m}$ and radius $\beta_{\rm
cir}^{m}$ encloses $\sigma^{m}$.

###### Proof.

The proof is obvious from Lemma 3.2. ∎

######  Remark 3.4.

In 1901 Jung121212 Heinrich Wilhelm Ewald Jung (1876 – 1953), German
mathematician. [42] was first answered to the question of best possible
estimate of the radius of the smallest spherical surface enclosing a bounded
subset $P$ of $\mathbb{R}^{n}$ of a given diameter, that is the maximal
distance of any two points of $P$. Particularly, Jung established results for
the case of finite point sets and indicated their extension to infinite sets.
In addition, in 1910 Jung [43] proposed necessary conditions on the smallest
circle enclosing a finite point set in a plane (for a discussion on these
issues and for a complete and elegant proof of Jung’s theorem, we refer the
interested reader to [10]).

######  Theorem 3.5 (Jung’s enclosing theorem (1901) [42]).

Assume that ${\rm diam}(P)$ is the diameter of a bounded subset ${P}$ of
$\mathbb{R}^{n}$ (containing more than a single point). Then, (a) there exists
a unique spherical surface of circumradius $\rho_{\rm cir}^{n}$ enclosing $P$,
and (b) $\rho_{\rm cir}^{n}\leqslant[n/(2n+2)]^{1/2}\,{\rm diam}(P)$.

###### Proof.

This theorem has received quite a few proofs due to Süss (1936) [73],
Blumenthal and Wahlin (1941) [10], Eggleston (1958) [28, Th.49, p.111] and
Guggenheimer (1977) [36, Pr.13-6, p.140], among others. ∎

In 1953 Gale [30] has provided the following equivalent formulation of Jung’s
theorem.

######  Theorem 3.6 (Gale’s enclosing theorem (1953) [30]).

The circumscribed $n$-sphere of a regular $n$-simplex of diameter 1 will cover
any $n$-dimensional set of diameter 1.

######  Lemma 3.7 (Regular simplex enclosing).

Let $\sigma^{m}$ be a regular $m$-simplex in ${\mathbb{R}}^{n}$ with edge
length ${\rm diam}(\sigma^{m})$, then the barycentric circumradius $\beta_{\rm
cir}^{m}$ is given by:

(3.3) $\beta_{\rm cir}^{m}=[m/(2m+2)]^{1/2}\,{\rm diam}(\sigma^{m}).$

###### Proof.

Since the lengths of the $m$ edges that have the common vertex
${\upsilon}^{i}$ as well as the lengths of the $\binom{m}{2}$ edges of the
$i$-th $(m-1)$-face $\sigma^{m}_{\neg i}$ are equal to ${\rm
diam}(\sigma^{m})$, by applying Lemma 3.2 we obtain

$\displaystyle\beta_{\rm cir}^{m}=$ $\displaystyle\frac{1}{m+1}\left[m^{2}{\rm
diam}(\sigma^{m})^{2}-\binom{m}{2}{\rm diam}(\sigma^{m})^{2}\right]^{1/2}$
$\displaystyle=$
$\displaystyle\frac{1}{m+1}\left[\frac{m(m+1)}{2}\right]^{1/2}{\rm
diam}(\sigma^{m})=\left[\frac{m}{2(m+1)}\right]^{1/2}{\rm diam}(\sigma^{m}).$

Thus the lemma is proved. ∎

The following corollary is related to Fermat’s problem (_cf_. Remark 2.26).

######  Corollary 3.8.

Assume that $\sigma^{m}=[{\upsilon}^{0},{\upsilon}^{1},\ldots,{\upsilon}^{m}]$
is an $m$-simplex in ${\mathbb{R}}^{n}$ $(n\geqslant m)$ with diameter ${\rm
diam}(\sigma^{m})$ and let $\kappa^{m}$ be its barycenter. Then it holds that

(3.4) $\sum_{i=0}^{m}\|\kappa^{m}-{\upsilon}^{i}\|=[{m(m+1)/2}]^{1/2}\,\,{\rm
diam}(\sigma^{m}).$

###### Proof.

The proof follows from Eq. (3.3). ∎

######  Remark 3.9.

In the case of regular simplices the barycentric circumradius and Jung’s
estimate are identical.

######  Remark 3.10.

The barycentric circumradius of an $m$-simplex $\sigma^{m}$ in
$\mathbb{R}^{n}$ with diameter ${\rm diam}(\sigma^{m})$ is not always greater
than the barycentric circumradius of any regular $m$-simplex $\tau^{m}$ in
$\mathbb{R}^{n}$ with the same diameter. Consider, for example, a regular
$2$-simplex $\tau^{2}=[{\upsilon}^{0},{\upsilon}^{1},{\upsilon}^{2}]$ with
diameter 1 and let $\kappa^{2}$ be its barycenter. By applying Lemma 3.2 we
obtain $\|\kappa^{2}-{\upsilon}^{i}\|=3^{-1/2}$ for $i=0,1,2.$ Thus the
barycentric circumradius of $\tau^{2}$ is $3^{-1/2}$. Consider now the
$2$-simplex $\sigma^{2}=[\kappa^{2},{\upsilon}^{1},{\upsilon}^{2}]$. The
lengths of the edges of $\sigma^{2}$ are $3^{-1/2}$, $3^{-1/2}$ and 1.
Therefore the diameter of $\sigma^{2}$ is 1. By applying Theorem 3.3 we find
that the barycentric circumradius of $\sigma^{2}$ is equal to $[7/27]^{1/2}$,
which is smaller than the estimation of the upper bound $3^{-1/2}$ for the
circumradius $\rho_{\rm cir}^{2}$ that is given by the Jung enclosing Theorem
3.5.

Based on the above we provide the following result.

######  Theorem 3.11 (Simplex enclosing).

Assume that $\sigma^{m}=[{\upsilon}^{0},{\upsilon}^{1},\ldots,{\upsilon}^{m}]$
is an $m$-simplex in ${\mathbb{R}}^{n}$ $(n\geqslant m)$ with diameter ${\rm
diam}(\sigma^{m})$ and let $\beta_{\rm cir}^{m}$ be its barycentric
circumradius. Then, there exists a unique spherical surface of circumradius
$\rho_{\rm cir}^{m}$ enclosing $\sigma^{m}$, and $\rho_{\rm
cir}^{m}\leqslant\min\bigl{\\{}\beta_{\rm cir}^{m},\,[m/(2m+2)]^{1/2}\,{\rm
diam}(\sigma^{m})\bigr{\\}}$.

###### Proof.

The proof is obvious from Theorem 3.3 and Theorem 3.5. ∎

######  Remark 3.12.

The computation of the barycentric circumradius does not require any
additional computational burden since it can be easily obtained during the
computation of the longest edge (diameter) of the simplex.

Next, the important theorems due to Carathéodory131313 Constantin Carathéodory
(1873 – 1950), Greek mathematician., Helly141414 Eduard Helly (1884 – 1943),
Austrian mathematician., Radon151515 Johann Karl August Radon (1887 – 1956),
Austrian mathematician., and Tverberg161616 Helge Arnulf Tverberg (1935 –
2020), Norwegian mathematician. are presented.

######  Theorem 3.13 (Carathéodory’s theorem (1907) [18]).

Any point in the convex hull of a finite point set in $\mathbb{R}^{n}$ is a
convex combination of some at most $n+1$ of these points.

######  Theorem 3.14 (Helly’s theorem (1913) [39]).

Let $C_{1},C_{2},\ldots,C_{k}$ be a finite family of convex subsets of
$\mathbb{R}^{n}$, with $k\geqslant n+1$. If the intersection of every $n+1$ of
these sets is nonempty, then the whole family has a nonempty intersection.

######  Theorem 3.15 (Radon’s theorem (1921) [64]).

Any set of $n+2$ points in $\mathbb{R}^{n}$ can be partitioned into two sets
whose convex hulls intersect.

######  Theorem 3.16 (Tverberg’s theorem (1966) [77]).

Every set with at least $(p-1)(n+1)+1$ points in $\mathbb{R}^{n}$ can be
partitioned into $p$ subsets whose convex hulls all have at least one point in
common.

######  Remark 3.17.

In 1970 Rockafellar171717 Ralph Tyrrell Rockafellar (b. 1935), American
mathematician. has pointed out that _“Carathéodory’s theorem is the
fundamental dimensionality result in convexity theory, and it is the source of
many other results in which dimensionality is prominent”_ [66]. In general,
the convex hull of a subset $P$ of $\mathbb{R}^{n}$ can be obtained by
carrying out all convex combinations of elements of $P$. On the other hand,
due to Carathéodory’s theorem it is not necessary to perform combinations
involving more than $n+1$ elements at a time.

######  Remark 3.18.

Although Theorem 3.14 was proposed by Helly in 1913, yet it was published in
1923 [39]. Radon’s theorem gives the partition of finite point sets so that
the convex hulls of the parts intersect. This partition is a special case of
the _Tverberg’s partition_. Tverberg gave the first proof of Theorem 3.16 in
1966, while in 1981 he gave a simpler proof [78]. Obviously, for $p=2$ we
obtain Radon’s theorem, while the case $n=2$ was proved by Birch181818 Bryan
John Birch (b. 1931), English mathematician. in 1959 [8].

######  Remark 3.19.

Theorems 3.13, 3.14 and 3.16 are equivalent in the sense that each one can be
deduced from another [24].

Based on the above Blumenthal and Wahlin in 1941 [10] have provided the
following result that permits the reduction of the enclosing problem to a
finite one concerning $n+1$ points.

######  Theorem 3.20 (Blumenthal-Wahlin theorem (1941) [10]).

If each set of $n+1$ points of a subset $P$ of $\mathbb{R}^{n}$ is enclosable
by a spherical surface of a given radius, then $P$ is itself enclosable by
this spherical surface.

######  Remark 3.21.

The barycentric circumradius $\beta_{\rm cir}^{n}=\beta_{\rm cir}^{n}(P)$ of a
subset $P$ of $\mathbb{R}^{n}$ is the supremum of the barycentric circumradii
of simplices with vertices in $P$. In the following theorem, an application of
the above theorems has shown that a bounded set $P$ can be covered by a
spherical surface of circumradius $\beta_{\rm cir}^{n}(P)$, which in many
cases gives a better result than Jung’s enclosing theorem (_cf._ Remark 3.10).

######  Theorem 3.22 (Variant of Jung’s enclosing theorem (1988) [81]).

Assume that ${\rm diam}(P)$ is the diameter of a bounded subset ${P}$ of
$\mathbb{R}^{n}$ (containing more than a single point) and let $\beta_{\rm
cir}^{n}=\beta_{\rm cir}^{n}(P)$ be its barycentric circumradius. Then, (a)
there exists a unique spherical surface of circumradius $\rho_{\rm cir}^{n}$
enclosing $P$, and (b) $\rho_{\rm cir}^{n}\leqslant\min\bigl{\\{}\beta_{\rm
cir}^{n}(P),\,[n/(2n+2)]^{1/2}\,{\rm diam}(P)\bigr{\\}}$.

####  3.1.1. Jung’s enclosing theorem and the minimum enclosing ball problem

The enclosing (covering) problem is related to the well known and widely
applied _minimum enclosing ball_ (MEB) problem, which refers to the
_determination of the unique spherical surface of smallest radius enclosing a
given bounded subset of $\mathbb{R}^{n}$._ The MEB problem is considered
significant in various issues of mathematics (_e.g_. geometry and topology,
computational and discrete geometry, non-Euclidean and convex geometry) with
applications including, among others, machine learning, computer graphics and
facility location for which a large amount of articles have been published.
This problem is also referred as _Euclidean 1-center_ , _minimax problem in
facility locations_ , _Chebyshev radius and Chebyshev center_ , _smallest
bounding ball_ , _minimum bounding sphere_ , _minimum spanning ball_ ,
_smallest enclosing sphere_ , _etc._

It is believed that the earliest known statement of the MEB problem was first
posed in _circa_ 300 B.C. by Euclid in his work _“Elements”_ and specifically
in Book IV, Proposition 5 which refers to circumscribe a circle about a given
triangle (_cf._ [29]). The general case of finding the smallest circle
enclosing a given finite set of points in $\mathbb{R}^{2}$ was first appeared
in 1857 by Sylvester191919 James Joseph Sylvester (1814 – 1897), English
mathematician. [74]. In addition, the earliest known procedure for tackling
the above case was also proposed in 1860 by Sylvester [75]. The same procedure
was independently proposed in 1885 by Chrystal202020 George Chrystal (1851 –
1911), Scottish mathematician. [20]. Thereafter, various algorithms for
determining the MEB of a set of points in the plane have been proposed.

It is worth mentioning that, the first optimal linear-time method for fixed
dimension has been presented in 1982 by Megiddo [52], (_cf_. [53]). In
addition, a simple and very fast in practice randomized algorithm to solve the
problem, also in $\mathbb{R}^{2}$ and $\mathbb{R}^{3}$, in expected linear
time has been proposed in 1991 by Welzl [89].

A relatively recent generalization of Welzl’s algorithm for point sets lying
in _information-geometric spaces_ as well as an interesting application for
solving a _statistical model estimation problem_ by computing the center of a
finite set of univariate normal distributions, have been given in 2008 by
Nielsen and Nock [60].

As we have already mentioned, Jung in 1901 was first answered to the question
of best possible estimate of the radius of a sphere in $\mathbb{R}^{n}$
enclosing a bounded subset of $\mathbb{R}^{n}$ of a given diameter (_cf_.
Theorem 3.5). Thereafter, the generalization of MEB in $\mathbb{R}^{n}$ has
been in the spotlight for many years and it remains an important issue of
study and further research in the field. It should be noted that, the solution
of this problem has been proved that is unique in the case of the _Euclidean
geometry_ (see, _e.g._ Welzl 1991 [89]), the _hyperbolic geometry_ (see,
_e.g._ Nielsen and Hadjeres 2015 [59]), the _Riemannian positive-definite
matrix manifold_ (see, _e.g._ Lang 1999 [47], Nielsen and Bhatia 2013 [58]),
and in any _Cartan-Hadamard manifold_ (see, _e.g._ Arnaudon and Nielsen 2013
[5]), _i.e._ Riemannian manifold that is complete and simply connected with
non-positive sectional curvatures (see, _e.g._ Nielsen 2020 [57]). Also, the
above solution is unique in any _Bruhat–Tits space_ (see, _e.g._ Lang 1999
[47]), _i.e._ complete metric space with a semi-parallelogram property, which
includes the Riemannian manifold of symmetric positive definite matrices (see,
_e.g._ Nielsen 2020 [57]). On the other hand, the solution of the MEB problem
may not be unique in a metric space, as for instance, in the case of _discrete
Hamming metric space_ (see, _e.g._ Mazumdar _et al._ 2013 [51]) which, in this
case, results to NP-hard computation (_i.e_. the complexity class of decision
problems that are intrinsically harder than those that can be solved by a
nondeterministic Turing machine in polynomial time) (see, _e.g._ Nielsen 2020
[57]). For various mathematical approaches to the MEB and related to it
problems, we refer the interest reader to [88].

####  3.1.2. Generalizations of Jung’s theorem to other spaces and geometries

In general, Jung’s enclosing theorem is considered as a cornerstone in the
field that strongly influencing later developments and generalizations to
other spaces and non-Euclidean geometries. A few examples are as follows.

A generalization of Jung’s theorem to _Minkowski spaces_ has been proved in
1938 by Bohnenblust [11]. A simpler proof of Bohnenblust’s result has been
given in 1955 by Leichtweiss [49]. The values of the _expansion constant_ and
_Jung’s constant_ determined by the geometric properties of a Minkowski space
have been studied in 1959 by Grünbaum [34]. Generalizations of Jung’s theorem
for a _pair of Minkowski spaces_ have been given in 2006 by Boltyanski and
Martini [14]. Particularly, they gave generalizations of Jung’s theorem for
the case where there are two Minkowski metrics in $\mathbb{R}^{n}$, one for
the diameter of a set $P\in\mathbb{R}^{n}$, and the other for the radius of
the Minkowski ball containing $P$. An one-to-one connection between the _Jung
constant_ determined by Jung’s theorem, _i.e._ the maximal ratio of the
circumradius and the diameter of a body, in an arbitrary Minkowski space and
the maximal _Minkowski asymmetry_ of the complete bodies within that space has
been stated in 2017 by Brandenberg and González Merino [16].

An _infinity dimensional Hilbert space_ extension of Jung’s theorem and its
application to _measures of noncompactness_ have been proposed in 1984 by
Danes̆ [23]. A complete characterization of the _extremal subsets of Hilbert
spaces_ , which constitutes an infinite-dimensional generalization of the Jung
theorem, has been proposed in 2006 by Nguen-Khac and Nguen-Van [56]. A
generalization of Jung’s theorem to convex regions on the _$n$ -dimensional
spherical surface_ has been proposed in 1946 by Santaló [70]. The
determination of the _Jung diameter_ of the _complex projective plane_ has be
given and lower bounds for all _complex projective spaces_ have been obtained
in 1985 by Katz [44]. A generalization of Jung’s theorem has been proved in
1985 by Dekster [25] for which estimates are given for the side-lengths of
certain inscribed simplices. The obtained result has been extended to the
_spherical and hyperbolic spaces_ by Dekster in 1995 [26] and to a class of
_metric spaces of curvature bounded above_ that includes _Riemannian spaces_
also by Dekster in 1997 [27].

An analogue of Jung’s theorem for _Alexandrov spaces of curvature bounded
above_ has been presented in 1997 by Lang and Schroeder [48]. Furthermore,
_combinatorial generalizations_ of Jung’s theorem have been presented and the
_“fractional”_ and _“colorful”_ versions of this theorem have been proved in
2013 by Akopyan [1].

###  3.2. Simplex thickness

######  Remark 3.23.

The thickness of a simplex is the dimensionless quantity that it is determined
by the ratio of the minimal distance from its barycenter to its boundary, to
that of the diameter of the simplex. In general, it provides a measure for the
quality or how well shaped a simplex is.

######  Lemma 3.24.

Let $\sigma^{m}$ be an $m$-simplex in ${\mathbb{R}}^{n}$ $(n\geqslant m)$ with
diameter ${\rm diam}(\sigma^{m})$ and boundary ${\vartheta}{\sigma}^{m}$.
Assume that $\kappa^{m}$ is its barycenter and let $\sigma^{m}_{\neg i}$ be
the $i$-th $(m-1)$-face of $\sigma^{m}$. Then, the barycentric inradius of
$\sigma^{m}$ is given by:

(3.5) $\beta_{\rm inr}^{m}=\min_{0\leqslant i\leqslant
m}\Bigl{\\{}\min_{x\in{\sigma_{\neg i}^{m}}}\|\kappa^{m}-x\|\Bigr{\\}},$

and the thickness of $\sigma^{m}$ is as folows:

(3.6) ${\theta}({\sigma}^{m})=\min_{0\leqslant i\leqslant
m}\Bigl{\\{}\min_{x\in{\sigma_{\neg i}^{m}}}\|\kappa^{m}-x\|\Bigr{\\}}/{\rm
diam}(\sigma^{m}).$

###### Proof.

Since it holds that ${\vartheta}\kern
0.56917pt$${\sigma}^{m}=\cup_{i=0}^{m}\sigma^{m}_{\neg i}$, the proof can be
easily obtained by companying Eq. (2.2) and Eq. (2.3). ∎

######  Lemma 3.25.

Assume that $\sigma^{m}=[{\upsilon}^{0},{\upsilon}^{1},\ldots,{\upsilon}^{m}]$
is an $m$-simplex in ${\mathbb{R}}^{n}$ $(n\geqslant m)$. Let $\kappa^{m}$ be
the barycenter of $\sigma^{m}$ and let $\kappa_{i}^{m}$ be the barycenter of
the $i$-th $(m-1)$-face $\sigma^{m}_{\neg i}$ of $\sigma^{m}$. Then for each
$i=0,1,\ldots,m$ it holds that:

(3.7)
$\|\kappa^{m}-\kappa^{m}_{i}\|=\frac{1}{m(m+1)}\Biggl{(}m\sum_{\begin{subarray}{c}j=0\\\
j\neq
i\end{subarray}}^{m}\|\upsilon^{i}-\upsilon^{j}\|^{2}-\sum_{\begin{subarray}{c}p=0\\\
p\neq i\end{subarray}}^{m-1}\sum_{\begin{subarray}{c}q=p+1\\\ q\neq
i\end{subarray}}^{m}\|\upsilon^{p}-\upsilon^{q}\|^{2}\Biggr{)}^{1/2}.$

###### Proof.

The proof can be easily obtained using Eq. (2.13) and Eq. (3.1). ∎

######  Definition 3.26.

The quantity

(3.8) $\tilde{\beta}_{\rm inr}^{m}=\min_{0\leqslant i\leqslant
m}\|\kappa^{m}-\kappa^{m}_{i}\|,$

is called _estimation of the barycentric inradius_ $\beta_{\rm inr}^{m}$ of
$\sigma^{m}$, while the quantity

(3.9) $\tilde{\theta}({\sigma}^{m})=\min_{0\leqslant i\leqslant
m}\|\kappa^{m}-\kappa^{m}_{i}\|/{\rm diam}(\sigma^{m}),$

is called _estimation of the thickness_ ${\theta}({\sigma}^{m})$ of
$\sigma^{m}$.

######  Remark 3.27.

Obviously, the quantities $\tilde{\beta}_{\rm inr}^{m}$ and
$\tilde{\theta}({\sigma}^{m})$ constitute upper bounds for the quantities
$\beta_{\rm inr}^{m}$ and ${\theta}({\sigma}^{m})$, respectively.

######  Lemma 3.28.

Assume that $\sigma^{m}$ is a regular $m$-simplex in ${\mathbb{R}}^{n}$ with
diameter ${\rm diam}(\sigma^{m})$, then the barycentric inradius $\beta_{\rm
inr}^{m}$ is given by:

(3.10) $\beta_{\rm inr}^{m}=\bigl{[}{2m(m+1)}\bigr{]}^{-1/2}{\rm
diam}(\sigma^{m}).$

###### Proof.

Since the lengths of the $m$ edges that have the common vertex
${\upsilon}^{i}$ as well as the lengths of the $\binom{m}{2}$ edges of the
$i$-th $(m-1)$-face $\sigma^{m}_{\neg i}$ are equal to ${\rm
diam}(\sigma^{m})$, by applying Lemma 3.25 we obtain

$\displaystyle\beta_{\rm inr}^{m}=$
$\displaystyle\frac{1}{m(m+1)}\left[m^{2}{\rm
diam}(\sigma^{m})^{2}-\binom{m}{2}{\rm diam}(\sigma^{m})^{2}\right]^{1/2}$
$\displaystyle=$
$\displaystyle\frac{1}{m(m+1)}\left[\frac{m(m+1)}{2}\right]^{1/2}{\rm
diam}(\sigma^{m})=\left[\frac{1}{2m(m+1)}\right]^{1/2}{\rm diam}(\sigma^{m}).$

Thus the lemma is proved. ∎

######  Remark 3.29.

The simplex thickness gives a _quality measure_ of a simplex. This is
important, among others, to piecewise linear approximations of smooth mappings
in fixed point theory and in complementarity theory. Also, it is significant
to simplicial and continuation methods for approximating fixed points as well
as in locating and computing periodic orbits of nonlinear mappings and
solutions of systems of nonlinear algebraic and/or transcendental equations.
In addition, in general, analysis of simplices characteristics in
$\mathbb{R}^{n}$ is useful in the implementation of high dimensional elements
of the well known and widely used finite element method that gives a formalism
for deriving discrete methods for approximating solutions of differential
equations (see, _e.g._ [6, 12, 13, 17, 46, 68, 83, 90]).

######  Definition 3.30 ([30]).

A set $P$ is said to be _inscribed_ in an $n$-simplex $\sigma^{n}$ in
$\mathbb{R}^{n}$ if $P\subset\sigma^{n}$ and $P$ intersects every $(n-1)$-face
of $\sigma^{n}$.

######  Remark 3.31.

The following important theorem states that: “ _The regular $n$-simplex whose
inscribed sphere has diameter 1 will cover any $n$-dimensional set of diameter
1_” [30, p.222].

######  Theorem 3.32 (Gale’s theorem (1953) [30]).

Assume that $P$ is a closed subset of $\mathbb{R}^{n}$ of diameter ${\rm
diam}(P)=1$. Then $P$ can be inscribed in a regular $n$-simplex $\sigma^{n}$
of diameter ${\rm diam}(\sigma^{n})\leqslant\bigl{[}n(n+1)/2\bigr{]}^{1/2}$.

######  Lemma 3.33.

The diameter of the regular $n$-simplex $\sigma^{n}$ in $\mathbb{R}^{n}$ whose
the inscribed sphere has diameter 1, is given by ${\rm
diam}(\sigma^{n})=\bigl{[}{n(n+1)/2}\bigr{]}^{1/2}$.

###### Proof.

Since the diameter of the inscribed sphere is $2\beta_{\rm inr}^{n}=1$, the
proof follows from Eq. (3.10). ∎

######  Remark 3.34.

Gale in [30, p.224] has pointed out that, since $\sqrt{n(n+1)/2}$ is the
diameter of the regular $n$-simplex whose inscribed sphere has diameter 1, the
inequality in Theorem 3.32 is the best possible. Also, he applied Theorem 3.32
to the case $n=2$ and he showed that every set in $\mathbb{R}^{2}$ of diameter
1 is the union of three sets each of diameter less than $\sqrt{3}/2<1$. This
result is related to the conjecture of Borsuk212121 Karol Borsuk (1905 –
1982), Polish mathematician. that states: “ _Every set in $\mathbb{R}^{n}$ of
diameter 1 is the union of $n+1$ sets each of diameter less than 1_” [15].

####  3.2.1. Relations between the circumradius, inradius, diameter and width

In 1958 Eggleston [28] pointed out that there are twelve possible inequalities
between any two of the four following characteristics of a convex set $P$ in
$\mathbb{R}^{n}$ (containing more than a single point).

Namely, the characteristics are: (a)  _circumradius_ $\rho_{\rm cir}^{n}$
(_i.e_. the radius of the smallest spherical surface enclosing the set), (b)
_inradius_ $\rho_{\rm inr}^{n}$ (_i.e_. the radius of the greatest spherical
surface which is contained in the set), (c)  _diameter_ ${\rm diam}(P)$
(_i.e_. the maximal distance of any two points of the set) and (d)  _width_
${\rm wid}(P)$ (_i.e_. the minimum distance of two parallel supporting
hyperplanes of the set).

The twelve possible inequalities are categorized as follows. Specifically, the
first four trivial inequalities are as follows:

(3.11a) $\displaystyle\rho_{\rm inr}^{n}$ $\displaystyle\leqslant\rho_{\rm
cir}^{n},$ $\displaystyle\rho_{\rm inr}^{n}$
$\displaystyle\leqslant\frac{1}{2}{\rm wid}(P),$ (3.11b) $\displaystyle{\rm
diam}(P)$ $\displaystyle\leqslant 2\rho_{\rm cir}^{n},$ $\displaystyle{\rm
wid}(P)$ $\displaystyle\leqslant{\rm diam}(P).$

The above inequalities imply the following two ones:

(3.12) $\kern 11.38092pt\rho_{\rm inr}^{n}\leqslant\frac{1}{2}{\rm
diam}(P),\kern 45.52458pt{\rm wid}(P)\leqslant 2\rho_{\rm cir}^{n}.$

Furthermore, Eggleston pointed out that since in the case where $\lambda$ is
not infinite there are no inequalities of the following four expressions:

(3.13a) $\displaystyle\rho_{\rm cir}^{n}$
$\displaystyle\leqslant\lambda\,\rho_{\rm inr}^{n},$ $\displaystyle\rho_{\rm
cir}^{n}$ $\displaystyle\leqslant\lambda\,{\rm wid}(P),$ (3.13b)
$\displaystyle{\rm diam}(P)$ $\displaystyle\leqslant\lambda\,\rho_{\rm
inr}^{n},$ $\displaystyle{\rm diam}(P)$ $\displaystyle\leqslant\lambda\,{\rm
wid}(P),$

and the remaining two inequalities are quite interesting and have the forms:

(3.14) $\kern 11.38092pt\rho_{\rm cir}^{n}\leqslant\lambda\,{\rm
diam}(P),\kern 45.52458pt{\rm wid}(P)\leqslant\lambda\,\rho_{\rm inr}^{n}.$

Particularly, the proof of the first of the above inequalities (3.14) that has
been given by Eggleston (_cf_. [28, Th. 49]) is related to _Jung’s theorem_
(_cf_. Theorem 3.5), while his proof of the second inequality (_cf_. [28, Th.
50]) is related to _Steinhagen’s theorem_ that it has been given in 1922 by
Steinhagen [71] (_cf_. Theorem 3.37).

In addition, it is worth mentioning that, in 1987
Perel${}^{\prime}\\!$man222222 Grigorii Yakovlevich Perel${}^{\prime}\\!$man
(b. 1966), Russian mathematician. in his publication _“ $k$ radii of a compact
convex body”_ [61] considered the _internal_ and _external_ $k$-radii of a
compact (_i.e_. closed and bounded) convex body $P$ in $\mathbb{R}^{n}$ that
are particular cases of the _diameters_ used in _Berns̆teĭn and Kolmogorov
approximation theory_ , respectively. Specifically, the _internal $k$-radius_
$r_{k}(P)$ for $k=1,2,\ldots,n$ of $P$ is defined as the radius of the
greatest $k$-dimensional sphere contained in $P$, while the _external
$k$-radius_ $R_{k}(P)$ for $k=1,2,\ldots,n$ of $P$ is defined as the smallest
radius of the solid cylinder containing $P$ with a $(n+1-k)$-dimensional
spherical cross section and a $(k-1)$-dimensional generator.

Perel${}^{\prime}\\!$man in [61] and independently Pukhov in [63] (_cf._ [32])
studied the relation between these internal and external $k$-radii measures,
and proved the following theorem:

######  Theorem 3.35 (Perel${}^{\prime}\\!$man-Pukhov theorem [61]).

Let $P$ be a compact convex body in $\mathbb{R}^{n}$, then it holds that:

(3.15) $\frac{R_{k}(P)}{r_{k}(P)}\leqslant k+1,\quad 1\leqslant k\leqslant n,$

where $R_{k}(P)$ and $r_{k}(P)$ are the external and internal $k$-radii of
$P$, respectively.

######  Remark 3.36.

Perel${}^{\prime}\\!$man in [61] pointed out, among others, that: (a) The
equality $r_{k}(P)=R_{k}(P)$ holds for certain bodies, _e.g_. spheres, (b)
Inequality (3.15) is also valid for finite $r_{k}(P)$ without the assumption
that the convex body $P$ is bounded, (c) The upper bound of Theorem 3.35 is
not a least upper bound for any $1\leqslant k\leqslant n$ and the estimate
given in Inequality (3.15) is extremely “rough” (non-sharp), (d) Precise
estimates of $R_{k}(P)/r_{k}(P)$ for the cases $k=2,3,\ldots,n-1$ are,
evidently, unknown, (e) For the cases $k=1$ and $k=n$, the maximum of
$R_{k}(P)/r_{k}(P)$ can be reached on a regular $n$-simplex, (f) Bounds which
are, in fact, least upper bounds for $R_{k}(P)/r_{k}(P)$ is the most
interesting, since the bound for $R_{\ell}(P)/r_{k}(P)$ for $k\leqslant\ell$
follows from it, (g) Least upper bounds for the cases $R_{1}(P)/r_{1}(P)$ and
$R_{d}(P)/r_{d}(P)$ can be obtained by Jung’s theorem [42] and the
generalization of the Blaschke232323 Wilhelm Johann Eugen Blaschke (1885 –
1962), Austrian mathematician. theorem [9], due to Steinhagen242424 Paul
Steinhagen, German mathematician, Dr. phil. Universität Hamburg 1920. [71],
respectively.

######  Theorem 3.37 (Steinhagen’s theorem (1922) [71]).

Let $P$ be a bounded convex subset of $\mathbb{R}^{n}$ of inradius $\rho_{\rm
inr}^{n}$. Then, the width ${\rm wid}(P)$ satisfies:

(3.16) ${{\rm wid}(P)}\leqslant\begin{cases}{2\sqrt{n}\,\rho_{\rm
inr}^{n},}&{\text{if}}\ {n}\,\ {\text{is odd}},\\\\[8.5359pt]
\bigl{[}{2(n+1)/\sqrt{n+2}\,\bigr{]}\,\rho_{\rm inr}^{n},}&{\text{if}}\ {n}\,\
{\text{is even}}.\end{cases}$

######  Remark 3.38.

The above statement of the theorem gives an upper bound for ${\rm wid}(P)$ in
terms of $\rho_{\rm inr}^{n}$ and we use this statement in order its
corresponding inequality to be compatible with the form of the above referred
secoond inequality in (3.14). The original statement of the theorem refers to
the lower bound for $\rho_{\rm inr}^{n}$ in terms of ${\rm wid}(P)$ and has
the following equivalent form (_cf_. [28, Th. 50]).

######  Theorem 3.39 (Steinhagen’s theorem (1922) [71]).

Let $P$ be a bounded convex subset of $\mathbb{R}^{n}$ of width ${\rm
wid}(P)$. Then, the inradius $\rho_{\rm inr}^{n}$ satisfies:

${\rho_{\rm
inr}^{n}}\geqslant\begin{cases}\bigl{(}2\sqrt{n}\bigr{)}^{-1}\,{{\rm
wid}(P)},&{\text{if}}\ {n}\,\ {\text{is odd}},\\\\[8.5359pt]
\bigl{[}\sqrt{n+2}/(2n+2)\bigr{]}\,{{\rm wid}(P)},&{\text{if}}\ {n}\,\
{\text{is even}}.\end{cases}$

######  Remark 3.40.

The Steinhagen theorem has been proved in 1914 for $n=2$ by Blaschke [9]. An
additional proof of Steinhagen’s theorem has been given in 1936 by Gericke
[31]. Furthermore, in 1946 Santaló [70] gave a _generalization of Jung’s and
Steinhagen’s theorems_ to convex regions on the _$n$ -dimensional spherical
surface_. A geometrical proof of the above results for $n=2$ has been given in
1944, also by Santaló [69]. It is worth noting that, Santaló’s result about
the generalization of the Steinhagen’s theorem, also generalizes the well-
known _Robinson’s theorem_ that it has been proved by Robinson in 1938 [65].
In 1992 Henk [40] by studying the relation between circumradius and diameter
of a convex body which has been proposed by Jung’s theorem, gave a natural
_generalization of Jung’s theorem_. Also, in 1993 Betke and Henk [7] by
analyzing the relation between inradius and width of a convex set that has
been established by Steinhagen’s theorem proposed respectively a natural
_generalization of Steinhagen’s theorem_.

######  Remark 3.41.

The circumradius, inradius and width of the regular $n$-simplex $\sigma^{n}$
in ${\mathbb{R}}^{n}$ with diameter 1 are given as follows. The circumradius
$\rho_{\rm cir}^{n}$ of $\sigma^{n}$, which in the case of regular simplices
is the same with the barycentric circumradius $\beta_{\rm cir}^{n}$ is
$\rho_{\rm cir}^{n}\equiv\beta_{\rm cir}^{n}=\sqrt{n/(2n+2)}$ (_cf_. Lemma 3.7
and Jung’s Theorem 3.5). Similarly, the inradius $\rho_{\rm inr}^{n}$ of
$\sigma^{n}$ which is the same with the barycentric inradius $\beta_{\rm
inr}^{n}$ is given by $\rho_{\rm inr}^{n}\equiv\beta_{\rm
inr}^{n}=1/\sqrt{2n(n+1)}$ (_cf_. Lemma 3.28). Also, the width of $\sigma^{n}$
is ${\rm wid}(\sigma^{n})\approx\sqrt{2/n}$ and specifically ${\rm
wid}(\sigma^{n})=\sqrt{2/(n+1)}$ if $n$ is odd and ${\rm
wid}(\sigma^{n})=\sqrt{2(n+1)}/\sqrt{n(n+2)}$ if $n$ is even (_cf_. [2, 33,
37]).

Based on Steinhagen’s theorem we give a proof for the above referred exact
bound on the width of the regular simplex in ${\mathbb{R}}^{n}$ with diameter
1.

######  Theorem 3.42 (Width of the regular simplex).

Let $\sigma^{n}$ be a regular $n$-simplex in ${\mathbb{R}}^{n}$ with diameter
${\rm diam}(\sigma^{n})=1$, then the width ${\rm wid}(\sigma^{n})$ of
$\sigma^{n}$ is given by:

(3.17) ${{\rm wid}(\sigma^{n})}=\begin{cases}{\sqrt{2/(n+1)},}&{\text{if}}\
{n}\,\ {\text{is odd}},\\\\[8.5359pt]
{\sqrt{2(n+1)}/\sqrt{n(n+2)},}&{\text{if}}\ {n}\,\ {\text{is
even}}.\end{cases}$

###### Proof.

The proof follows from Lemma 3.28 and Theorem 3.37. The equality holds since
$\sigma^{n}$ is regular (_cf_. [2] and Remark 2.27). ∎

###  3.3. Simplex bisection

######  Remark 3.43.

An $m$-simplex of $\mathbb{R}^{n}$ is bisected by choosing its longest edge,
dividing it in two equal parts and defining two new $m$-simplices. Each of
them will, in turn, be bisected and this process continues for a finite number
of iterations.

######  Definition 3.44 (Simplex bisection [45, 67, 72]).

Let
${\sigma}^{m}_{\langle{0}\rangle}=[{\upsilon}^{0},{\upsilon}^{1},\ldots,{\upsilon}^{m}]$
be an $m$-simplex in $\mathbb{R}^{n}$ $(n\geqslant m)$. Assume that
$[{\upsilon}^{i},{\upsilon}^{j}]$ is the longest edge of
${\sigma}^{m}_{\langle{0}\rangle}$ and let
$M=({\upsilon}^{i}+{\upsilon}^{j})/2$ be its midpoint. Then the bisection of
${\sigma}^{m}_{\langle{0}\rangle}$ is the order pair of $m$-simplices
$\bigl{\\{}{{\sigma}^{m}_{\langle{10}\rangle},\,{\sigma}^{m}_{\langle{11}\rangle}}\bigr{\\}}$
where:

$\displaystyle{\sigma}^{m}_{\langle{10}\rangle}=[{\upsilon}^{0},{\upsilon}^{1},\ldots,{\upsilon}^{i-1},M,{\upsilon}^{i+1},\ldots,{\upsilon}^{j},\ldots,{\upsilon}^{m}],$
$\displaystyle{\sigma}^{m}_{\langle{11}\rangle}=[{\upsilon}^{0},{\upsilon}^{1},\ldots,{\upsilon}^{i},\ldots,{\upsilon}^{j-1},M,{\upsilon}^{j+1},\ldots,{\upsilon}^{m}].$

The $m$-simplices ${\sigma}^{m}_{\langle{10}\rangle}$ and
${\sigma}^{m}_{\langle{11}\rangle}$ are called _lower simplex_ and _upper
simplex_ respectively corresponding to ${\sigma}^{m}_{\langle{0}\rangle}$,
while both ${\sigma}^{m}_{\langle{10}\rangle}$ and
${\sigma}^{m}_{\langle{11}\rangle}$ are called _elements of the bisection_ of
${\sigma}^{m}_{\langle{0}\rangle}$. Next, each of the elements of the
bisection will, in turn, be bisected and the process will be continued to get
a sequence of sets of simplices. In case where one of the elements is selected
then this element is called _selected $m$-simplex produced after one bisection
of_ ${\sigma}^{m}_{\langle{0}\rangle}$ and it is denoted by
${\sigma}^{m}_{\langle{1}\rangle}$. Assume now that the bisection is applied
with ${\sigma}^{m}_{\langle{1}\rangle}$ replacing
${\sigma}^{m}_{\langle{0}\rangle}$ giving thus the
${\sigma}^{m}_{\langle{2}\rangle}$. Suppose further that this process
continues for $p$ iterations. Then the ${\sigma}^{m}_{\langle{p}\rangle}$ is
called the _selected $m$-simplex produced after $p$ iterations of the
bisection of ${\sigma}^{m}_{\langle{0}\rangle}$_.

######  Theorem 3.45 (Kearfott’s theorem (1978) [45]).

Let
${\sigma}^{m}_{\langle{0}\rangle}=[{\upsilon}^{0},{\upsilon}^{1},\ldots,{\upsilon}^{m}]$
be an $m$-simplex in $\mathbb{R}^{n}$ $(n\geqslant m)$ and let
${\sigma}^{m}_{\langle{p}\rangle}$ be any $m$-simplex produced after $p$
bisections of ${\sigma}^{m}_{\langle{0}\rangle}$. Then it holds that:

(3.18) ${\rm
diam}\bigl{(}{\sigma}^{m}_{\langle{p}\rangle}\bigr{)}\leqslant\bigl{(}\sqrt{3}/2\bigr{)}^{\lfloor
p/m\rfloor}{\rm diam}\bigl{(}{\sigma}^{m}_{\langle{0}\rangle}\bigr{)},$

where ${\rm diam}\bigl{(}{\sigma}^{m}_{\langle{p}\rangle}\bigr{)}$ and ${\rm
diam}\bigl{(}{\sigma}^{m}_{\langle{0}\rangle}\bigr{)}$ are the diameters of
${\sigma}^{m}_{\langle{p}\rangle}$ and ${\sigma}^{m}_{\langle{0}\rangle}$
respectively and $\lfloor p/m\rfloor$ is the largest integer less than or
equal to $p/m$.

######  Lemma 3.46.

Assume that ${\sigma}^{m}_{\langle{0}\rangle}$,
${\sigma}^{m}_{\langle{p}\rangle}$, ${\rm
diam}\bigl{(}{\sigma}^{m}_{\langle{0}\rangle}\bigr{)}$ and ${\rm
diam}\bigl{(}{\sigma}^{m}_{\langle{p}\rangle}\bigr{)}$ are as in Theorem 3.45.
Let $\kappa^{m}_{\langle{p}\rangle}$ and $\beta_{\rm cir}^{m}=\beta_{\rm
cir}^{m}\bigl{(}{\sigma}^{m}_{\langle{p}\rangle}\bigr{)}$ be the barycenter
and the barycentric circumradius of ${\sigma}^{m}_{\langle{p}\rangle}$
respectively. Then for any point $x$ in ${\sigma}^{m}_{\langle{p}\rangle}$ it
holds that:

(3.19)
$\|x-\kappa^{m}_{\langle{p}\rangle}\|\leqslant\frac{m}{m+1}\bigl{(}\sqrt{3}/2\bigr{)}^{\lfloor
p/m\rfloor}{\rm diam}\bigl{(}{\sigma}^{m}_{\langle{0}\rangle}\bigr{)}.$

###### Proof.

By Theorem 3.3 we obtain that

$\|x-\kappa^{m}_{\langle{p}\rangle}\|\leqslant\beta_{\rm
cir}^{m}\bigl{(}{\sigma}^{m}_{\langle{p}\rangle}\bigr{)}\leqslant\frac{1}{m+1}\Bigl{(}m^{2}\,{\rm
diam}\bigl{(}{\sigma}^{m}_{\langle{p}\rangle}\bigr{)}^{2}\Bigr{)}^{1/2},$

and the result follows by Theorem 3.45. ∎

######  Lemma 3.47.

Assume that ${\sigma}^{m}_{\langle{0}\rangle}$,
${\sigma}^{m}_{\langle{p}\rangle}$, ${\rm
diam}\bigl{(}{\sigma}^{m}_{\langle{0}\rangle}\bigr{)}$ and ${\rm
diam}\bigl{(}{\sigma}^{m}_{\langle{p}\rangle}\bigr{)}$ are as in Theorem 3.45.
Assume further that ${\rm
shor}\bigl{(}{\sigma}^{m}_{\langle{p}\rangle}\bigr{)}$ is the length of the
shortest edge of ${\sigma}^{m}_{\langle{p}\rangle}$ and let
$\kappa^{m}_{\langle{p}\rangle}$ and $\beta_{\rm cir}^{m}=\beta_{\rm
cir}^{m}\bigl{(}{\sigma}^{m}_{\langle{p}\rangle}\bigr{)}$ be the barycenter
and the barycentric circumradius of ${\sigma}^{m}_{\langle{p}\rangle}$
respectively. Then for any point $x$ in ${\sigma}^{m}_{\langle{p}\rangle}$ it
holds that:

(3.20) $\|x-\kappa^{m}_{\langle{p}\rangle}\|\leqslant\frac{m}{m+1}\Bigl{(}{\rm
diam}\bigl{(}{\sigma}^{m}_{\langle{p}\rangle}\bigr{)}^{2}-\frac{m-1}{2m}\,{\rm
shor}\bigl{(}{\sigma}^{m}_{\langle{p}\rangle}\bigr{)}^{2}\Bigr{)}^{1/2}.$

###### Proof.

By Theorem 3.3 we obtain that

$\displaystyle\|x-\kappa^{m}_{\langle{p}\rangle}\|$
$\displaystyle\leqslant\beta_{\rm
cir}^{m}\bigl{(}{\sigma}^{m}_{\langle{p}\rangle}\bigr{)}\leqslant\frac{1}{m+1}\Bigl{(}m^{2}\,{\rm
diam}\bigl{(}{\sigma}^{m}_{\langle{p}\rangle}\bigr{)}^{2}-\binom{m}{2}\,{\rm
shor}\bigl{(}{\sigma}^{m}_{\langle{p}\rangle}\bigr{)}^{2}\Bigr{)}^{1/2}$
$\displaystyle\leqslant\frac{1}{m+1}\Bigl{(}m^{2}\,{\rm
diam}\bigl{(}{\sigma}^{m}_{\langle{p}\rangle}\bigr{)}^{2}-\frac{m(m-1)}{2}\,{\rm
shor}\bigl{(}{\sigma}^{m}_{\langle{p}\rangle}\bigr{)}^{2}\Bigr{)}^{1/2}.$

Thus the lemma is proved. ∎

######  Remark 3.48.

Simplex bisection methods are used for the computation of solutions of systems
of nonlinear algebraic and/or transcendental equations of the form:

(3.21) $F_{n}(x)=\mathit{\Theta^{n}},\,\,\mbox{ with
}\,\,F_{n}=(f_{1},f_{2},\ldots,f_{n}):{\mathcal{D}}\subset\mathbb{R}^{n}\to\mathbb{R}^{n},$

where $\mathit{\Theta^{n}}=(0,0,\ldots,0)$ is the origin of $\mathbb{R}^{n}$
and ${\mathcal{D}}$ is a bounded domain. Specifically, an $n$-simplex of
$\mathbb{R}^{n}$ can be bisected by choosing its longest edge, dividing it in
two equal parts and considering two new $n$-simplices. Each of them or one of
them is, in turn, bisected. This procedure continues for a finite number of
iterations for obtaining an approximation $s^{*}$ of a solution $s$ of Eq.
(3.21). For the selection of an element of the bisection process in each
iteration a criterion is applied for the existence of a solution of Eq. (3.21)
in this element (see, _e.g._ [38, 50, 54, 72, 79, 80, 82, 83, 84, 85, 86,
87]). In general, the bisection methods require only the algebraic signs of
the function values and are of major importance for solving problems with
imprecise information. Notice that the solution of Eq. (3.21) is equivalent to
the problem of computing fixed points of continuous functions in several
variables as well as periodic orbits of nonlinear mappings and similarly,
fixed points of the Poincaré map on a surface of section (see, _e.g._ [83,
87]).

######  Definition 3.49.

Assume that $\sigma^{n}$ is an $n$-simplex in $\mathbb{R}^{n}$ and let ${\rm
diam}(\sigma^{n})$ and ${\rm shor}(\sigma^{n})$ be the diameter and the length
of the shortest edge of $\sigma^{n}$ respectively. Assume further that $s$ is
a solution of Eq. (3.21) in $\sigma^{n}$. Then the barycenter $\kappa^{n}$ of
$\sigma^{n}$ is defined to be an _approximation_ of $s$ and the quantity:

(3.22) $\varepsilon^{n}=\frac{n}{n+1}\Bigl{(}{\rm
diam}(\sigma^{n})^{2}-\frac{n-1}{2n}\,{\rm
shor}(\sigma^{n})^{2}\Bigr{)}^{1/2},$

to be an _error estimate_ for $\kappa^{n}$.

######  Lemma 3.50.

Let ${\sigma}^{n}_{\langle{p}\rangle}$ be the selected $n$-simplex produced
after $p$ bisections of an $n$-simplex ${\sigma}^{n}_{\langle{0}\rangle}$ in
$\mathbb{R}^{n}$ with diameter ${\rm
diam}\bigl{(}{\sigma}^{m}_{\langle{0}\rangle}\bigr{)}$. Let ${\rm
shor}\bigl{(}{\sigma}^{m}_{\langle{p}\rangle}\bigr{)}$ be the length of the
shortest edge of ${\sigma}^{m}_{\langle{p}\rangle}$ and ${\rm
diam}\bigl{(}{\sigma}^{m}_{\langle{p}\rangle}\bigr{)}$ its diameter. Then

$\varepsilon^{n}_{\langle{p}\rangle}\leqslant\frac{n}{n+1}\bigl{(}\sqrt{3}/2\bigr{)}^{\lfloor
p/n\rfloor}{\rm diam}\bigl{(}{\sigma}^{n}_{\langle{0}\rangle}\bigr{)}.$

###### Proof.

Using Eq. (3.22) the proof follows by Lemmata 3.46 and 3.47. ∎

######  Theorem 3.51 (Convergence of simplex bisection (1986) [79]).

Let ${\sigma}^{n}_{\langle{p}\rangle}$ be the selected $n$-simplex produced
after $p$ bisections of an $n$-simplex ${\sigma}^{n}_{\langle{0}\rangle}$ in
$\mathbb{R}^{n}$. Assume that $s$ is a solution of Eq. (3.21) in
${\sigma}^{n}_{\langle{p}\rangle}$ and let $\kappa^{n}_{\langle{p}\rangle}$
and $\varepsilon^{n}_{\langle{p}\rangle}$ be the approximation of $s$ and the
error estimate for $\kappa^{n}_{\langle{p}\rangle}$ respectively. Then
$\lim_{p\to\infty}\varepsilon^{n}_{\langle{p}\rangle}=0$ and
$\lim_{p\to\infty}\kappa^{n}_{\langle{p}\rangle}=s.$

###### Proof.

Using Eq. (3.22) the proof follows by Lemma 3.46. ∎

######  Remark 3.52.

A significant related problem (which still remains attractive) refers to the
determination of a positive bound on the ratio of the lengths of the smallest
edges of simplices to the lengths of the largest edges, as the simplex
bisection is proceed [45].

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