# On separability in discrete geometry

Károly Bezdek and Zsolt Lángi  Károly Bezdek, Department of Mathematics and
Statistics, University of Calgary, Canada, and Department of Mathematics,
University of Pannonia, Veszprém, Hungary<EMAIL_ADDRESS>Zsolt
Lángi, Department of Algebra and Geometry, Budapest University of Technology
and Economics, Hungary, and Alfréd Rényi Institute of Mathematics, Budapest,
Hungary<EMAIL_ADDRESS>
###### Abstract.

A problem of Erdős ([GoGo45]) and a theorem of Fejes Tóth and Fejes Tóth
([FeFe]) initiated the study of non-separable arrangements of convex bodies
and the investigation of totally separable packings of convex bodies with both
topics analyzing the concept of separability from the point view of discrete
geometry. This article surveys the progress made on these and some closely
related problems and highlights the relevant questions that have been left
open.

###### Key words and phrases:

Non-separable arrangement, successively non-separable family, convex body,
positive homothetic copy, spherical cap, totally separable packing, density,
locally separable sphere packing, contact graph, contact number.

###### 2010 Mathematics Subject Classification:

52-02, 52A20, 52A38, 52A40, 52C07, 52C10, 52C17, 52C45

##  1\. Introduction

Separation by hyperplanes is a basic and often used concept in convex geometry
([Sch14]). So, perhaps not surprisingly separability properties of families of
convex sets play an important role in discrete geometry as well. In
particular, there are a number of recent results that have been discovered in
response to the questions about non-separable arrangements (resp., totally
separable packings) raised by Erdős [GoGo45] (resp., Fejes Tóth and Fejes Tóth
[FeFe]). Thus, we felt that it is timely to survey them and orient the readers
attention to those questions that have been left open and seem to have the
potential to motivate further research. It is convenient as well as natural to
group the results into three sections, each representing a topic of
independent interest. Section 2 discusses the state of the art of minimal
coverings of non-separable arrangements of convex bodies (resp., spherical
caps) in Euclidean spaces (resp., spherical spaces). Section 3 surveys dense
totally separable translative packings of convex bodies in Euclidean spaces
and densest totally separable packings of congruent spherical caps in the
spherical plane. Finally, Section 4 gives an overview on bounds for contact
numbers of locally separable sphere packings in Euclidean spaces.

Throughout, we use some standard notations and concepts from convex geometry
([Sch14]) as well as discrete geometry ([Bez10]). This includes the following
basic ones. The Euclidean norm of a vector (i.e., a point) $\mathbf{p}$ in the
$d$-dimensional Euclidean space ${\mathbb{E}}^{d}$ is denoted by
$|\mathbf{p}|:=\sqrt{\langle\mathbf{p},\mathbf{p}\rangle}$, where
$\langle\cdot,\cdot\rangle$ stands for the standard inner product. We denote
the $(d-1)$-dimensional unit sphere centered at the origin
$\mathbf{o}\in{\mathbb{E}}^{d}$ by
${\mathbb{S}}^{d-1}:=\\{\mathbf{u}\in{\mathbb{E}}^{d}\;:\;|\mathbf{u}|=1\\}$.
We shall use ${\rm vol}_{d}(\cdot)$ for the $d$-dimensional volume (i.e.,
Lebesgue measure) of sets in ${\mathbb{E}}^{d}$ and use ${\rm area}(\cdot)$
instead of ${\rm vol}_{2}(\cdot)$ in ${\mathbb{E}}^{2}$. The closed
$d$-dimensional Euclidean ball of radius $\rho>0$ centered at
$\mathbf{p}\in{\mathbb{E}}^{d}$ is denoted by
$\mathbf{B}[\mathbf{p},\rho]:=\\{\mathbf{q}\in{\mathbb{E}}^{d}\;:\;|\mathbf{p}-\mathbf{q}|\leq\rho\\}$
with ${\rm vol}_{d}(\mathbf{B}[\mathbf{p},\rho])=\rho^{d}\kappa_{d}$, where
$\kappa_{d}:={\rm vol}_{d}\left({\mathbf{B}[\mathbf{o},1]}\right)$. Moreover,
${\rm bd}(\cdot),{\rm int}(\cdot)$, and ${\rm conv}(\cdot)$ refer to the
boundary, interior, and the convex hull, respectively, of the corresponding
sets. Furthermore, let $R(\mathbf{C})$ (resp., $r(\mathbf{C})$) denote the
circumradius (resp., inradius) of the compact set $\mathbf{C}$ with non-empty
interior in $\mathbb{E}^{d}$, which is the radius of the smallest (resp., a
largest) $d$-dimensional ball that contains (resp., is contained in)
$\mathbf{C}$.

##  2\. Minimal coverings of non-separable arrangements

###  2.1. Non-separable arrangements in Euclidean spaces

The following theorem was conjectured by Erdős and proved by Goodman and
Goodman in [GoGo45].

######  Theorem 2.1.

Let the disks
$\mathbf{B}[\mathbf{x}_{1},\tau_{1}]\subset{\mathbb{E}}^{2},\dots,\mathbf{B}[\mathbf{x}_{n},\tau_{n}]\subset{\mathbb{E}}^{2}$
have the following property: No line of ${\mathbb{E}}^{2}$ divides the disks
$\mathbf{B}[\mathbf{x}_{1},\tau_{1}],\dots,\mathbf{B}[\mathbf{x}_{n},\tau_{n}]$
into two non-empty families without touching or intersecting at least one
disk. Then the disks
$\mathbf{B}[\mathbf{x}_{1},\tau_{1}],\dots,\mathbf{B}[\mathbf{x}_{n},\tau_{n}]$
can be covered by a disk of radius $\tau:=\sum_{i=1}^{n}\tau_{i}$.

The proof of Goodman and Goodman [GoGo45] is based on the following statement.

######  Lemma 2.2.

Let
$\mathcal{F}=\left\\{[x_{i}-\tau_{i},x_{i}+\tau_{i}]:\tau_{i}>0,i=1,2,\ldots,n\right\\}$
be a family of closed intervals in ${\mathbb{R}}$ such that
$\bigcup\mathcal{F}$ is single closed interval in ${\mathbb{R}}$. Let
$x:=\frac{\sum_{i=1}^{n}\tau_{i}x_{i}}{\sum_{i=1}^{n}\tau_{i}}$. Then the
interval $\left[x-\sum_{i=1}^{n}\tau_{i},x+\sum_{i=1}^{n}\tau_{i}\right]$
covers $\bigcup\mathcal{F}$.

With the help of Lemma 2.2 it is shown in [GoGo45] that
$\bigcup_{i=1}^{n}\mathbf{B}[\mathbf{x}_{i},\tau_{i}]\subset\mathbf{B}\left[\frac{\sum_{i=1}^{n}\tau_{i}\mathbf{x}_{i}}{\sum_{i=1}^{n}\tau_{i}},\sum_{i=1}^{n}\tau_{i}\right]$,
finishing the proof of Theorem 2.1.

For a completely different proof of Theorem 2.1 see Theorem 6.1 of Bezdek and
Litvak [BeLi16].

Notice that the value of $\tau$ in Theorem 2.1 (as a function of
$\tau_{1},\dots,\tau_{n}$) is the smallest possible. Namely, if the centers of
the disks $\mathbf{B}_{1},\dots,\mathbf{B}_{n}$ lie on a line such that the
consecutive disks touch each other, then the radius of the smallest disk
containing them is equal to $\sum_{i=1}^{n}\tau_{i}$.

Recall that a convex domain of ${\mathbb{E}}^{2}$ is a compact convex set with
non-empty interior. Hadwiger [Ha47] extended Theorem 2.1 by introducing the
concept of non-separable system as follows: A system of convex domains
$\mathbf{K}_{i}\subset{\mathbb{E}}^{2}$, $i=1,\dots,n$ is called separable if
there is a line of ${\mathbb{E}}^{2}$, which is disjoint from each
$\mathbf{K}_{i}$ and which divides ${\mathbb{E}}^{2}$ into two open half
planes each containing at least one $\mathbf{K}_{i}$. In the opposite case we
call the system non-separable. Projections on lines and Cauchy’s perimeter
formula combined with Theorem 2.1 yield the following inequalities.

######  Theorem 2.3.

Let $\mathbf{K}_{i}$, $i=1,\dots,n$ be a non-separable system of convex
domains in ${\mathbb{E}}^{2}$. If $\mathbf{K}_{0}:={\rm
conv}\left(\cup_{i=1}^{n}\mathbf{K}_{i}\right)$ and ${\rm
per}\left(\mathbf{K}_{i}\right),{\rm diam}\left(\mathbf{K}_{i}\right)$, and
$R\left(\mathbf{K}_{i}\right)$ denote the perimeter, the diameter, and the
circumradius, respectively, of the convex domain $\mathbf{K}_{i}$,
$i=0,1,\dots,n$, then

(1) ${\rm per}\left(\mathbf{K}_{0}\right)\leq\sum_{i=1}^{n}{\rm
per}\left(\mathbf{K}_{i}\right),{\rm
diam}\left(\mathbf{K}_{0}\right)\leq\sum_{i=1}^{n}{\rm
diam}\left(\mathbf{K}_{i}\right),\ {\rm and}\
R\left(\mathbf{K}_{0}\right)\leq\sum_{i=1}^{n}R\left(\mathbf{K}_{i}\right).$

It is natural to extend the problem of Erdős to higher dimensions as follows.

######  Definition 1.

Let $\mathbf{K}$ be a convex body in ${\mathbb{E}}^{d}$ and let
$\mathcal{K}:=\\{\mathbf{x}_{i}+\tau_{i}\mathbf{K}\ |\
\mathbf{x}_{i}\in{\mathbb{R}}^{d},\tau_{i}>0,i=1,2,\ldots,n\\}$, where $d\geq
2$ and $n\geq 2$. Assume that $\mathcal{K}$ is a non-separable family in
short, an NS-family, meaning that every hyperplane intersecting ${\rm
conv}\left(\bigcup\mathcal{K}\right)$ intersects a member of $\mathcal{K}$ in
${\mathbb{R}}^{d}$, i.e., there is no hyperplane disjoint from
$\bigcup\mathcal{K}$ that strictly separates some elements of $\mathcal{K}$
from all the other elements of $\mathcal{K}$ in ${\mathbb{E}}^{d}$. Then, let
$\lambda(\mathcal{K})>0$ denote the smallest positive value $\lambda$ such
that a translate of
$\lambda\left(\sum\limits_{i=1}^{n}\tau_{i}\right)\mathbf{K}$ covers
$\bigcup\mathcal{K}$.

In terms of Definition 1, Theorem 2.2 states that if $\mathcal{C}$ is an
arbitrary NS-family of finitely many homothetic ellipses in
${\mathbb{E}}^{2}$, then $\lambda(\mathcal{C})\leq 1$. In [BezLan16], the
authors showed that Theorem 2.1 actually holds not only for balls, but also
for any centrally-symmetric convex body. In other words, Theorem 2.1 extends
to balls of finite dimensional normed spaces.

######  Theorem 2.4.

For every $\mathbf{o}$-symmetric convex body $\mathbf{K}_{0}$ and every NS-
family $\mathcal{K}:=\\{\mathbf{x}_{i}+\tau_{i}\mathbf{K}_{0}\ |\
\mathbf{x}_{i}\in{\mathbb{E}}^{d},\tau_{i}>0,i=1,2,\ldots,n\\}$ the inequality
$\lambda(\mathcal{K})\leq 1$ holds for all $d\geq 2$ and $n\geq 2$.

The proof of Theorem 2.4 in [BezLan16] is a short one, which we recall next.
Let
$\mathbf{x}=\frac{\sum_{i=1}^{n}\tau_{i}\mathbf{x}_{i}}{\sum_{i=1}^{n}\tau_{i}}$,
and set
$\mathbf{K}^{\prime}=\mathbf{x}+\left(\sum_{i=1}^{n}\tau_{i}\right)\mathbf{K}_{0}$.
We prove that $\mathbf{K}^{\prime}$ covers $\bigcup\mathcal{K}$. For any line
$L$ through the origin $\mathbf{o}$, let
$\operatorname{proj}_{L}:{\mathbb{R}}^{d}\to L$ denote the orthogonal
projection onto $L$, and let $h_{\mathcal{K}}:\mathbb{S}^{d-1}\to{\mathbb{R}}$
and $h_{\mathbf{K}^{\prime}}:\mathbb{S}^{d-1}\to{\mathbb{R}}$ denote the
support functions of ${\rm conv}\left(\bigcup\mathcal{K}\right)$ and
$\mathbf{K}^{\prime}$, respectively. Then
$\operatorname{proj}_{L}\left(\bigcup\mathcal{K}\right)$ is a single interval,
which, by Lemma 2.2, is covered by
$\operatorname{proj}_{L}\left(\mathbf{K}^{\prime}\right)$. Thus, for any
$\mathbf{u}\in\mathbb{S}^{d-1}$, we have that $h_{\mathcal{K}}(\mathbf{u})\leq
h_{\mathbf{K}^{\prime}}(\mathbf{u})$, which readily implies that
$\bigcup\mathcal{K}\subseteq\mathbf{K}^{\prime}$.

Perhaps, it is not surprising that Goodman and Goodman [GoGo45] put forward
the following conjecture: For every convex body $\mathbf{K}$ in
${\mathbb{E}}^{d}$ and every NS-family
$\mathcal{K}:=\\{\mathbf{x}_{i}+\tau_{i}\mathbf{K}\ |\
\mathbf{x}_{i}\in{\mathbb{E}}^{d},\tau_{i}>0,i=1,2,\ldots,n\\}$ the inequality
$\lambda(\mathcal{K})\leq 1$ holds for all $d\geq 2$ and $n\geq 2$. However,
the authors [BezLan16] found the following counterexample to it.

Figure 1. A counterexample in the plane for three triangles.

Place three regular triangles
${\mathcal{T}}=\\{\mathbf{T}_{1},\mathbf{T}_{2},\mathbf{T}_{3}\\}$ of unit
side lengths into a regular triangle $\mathbf{T}$ of side length
$2+\frac{2}{\sqrt{3}}=3.154700\ldots>3$ such that

  * •

each side of $\mathbf{T}$ contains a side of $\mathbf{T}_{i}$, for $i=1,2,3$,
respectively (cf. Figure 1),

  * •

for $i=1,2,3$, the vertices of $\mathbf{T}_{i}$ contained in a side of
$\mathbf{T}$ divide this side into three segments of lengths
$\frac{2}{3}+\frac{1}{\sqrt{3}}$, $1$, and $\frac{1}{3}+\frac{1}{\sqrt{3}}$,
in counter-clockwise order.

A simple computation yields that the convex hull of any two of the small
triangles touches the third triangle, and hence, no two of them can be
strictly separated from the third one. Thus,
$\lambda({\mathcal{T}})=\frac{2}{3}+\frac{2}{3\sqrt{3}}=1.0515\ldots>1$ and
${\mathcal{T}}$ is a counterexample to the above conjecture of Goodman and
Goodman for $n=3$ in ${\mathbb{E}}^{2}$. Note that for any value $n\geq 4$,
placing $n-3$ sufficiently small triangles inside $\mathbf{T}$ yields a
counterexample for $n$ positive homothetic copies of a triangle in
${\mathbb{E}}^{2}$. As in [BezLan16], for $d\geq 2$ let $\lambda_{\rm
simplex}^{(d)}$ denote the supremum of $\lambda(\mathcal{K})$, where
$\mathcal{K}$ runs over the NS-families of finitely many positive homothetic
$d$-simplices in ${\mathbb{E}}^{d}$. It was observed in [BezLan16] that
$\lambda_{\rm simplex}^{(d)}$ is a non-decreasing sequence of $d$. Moreover,
Figure 2 shows how to extend the configuration in Figure 1 to
${\mathbb{E}}^{3}$, implying that $\lambda_{\rm simplex}^{(d)}\geq\lambda_{\rm
simplex}^{(2)}\geq\frac{2}{3}+\frac{2}{3\sqrt{3}}=1.0515\ldots>1$ for all
$d\geq 3$. The special role of $\lambda_{\rm simplex}^{(d)}$ is highlighted by
the following statement proved by the authors in [BezLan16].

Figure 2. A counterexample in ${\mathbb{R}}^{3}$ for three tetrahedra. The
dotted lines denote the intersection of a plane touching all three teterahedra
with the smallest tetrahedron containing them.

######  Lemma 2.5.

$\lambda_{\rm simplex}^{(d)}=\sup_{\mathcal{K}}\lambda(\mathcal{K})$ for all
$d\geq 2$, where $\mathcal{K}$ runs over the NS-families of finitely many
positive homothetic copies of an arbitrary convex body $\mathbf{K}$ in
${\mathbb{E}}^{d}$.

As the proof of Lemma 2.5 in [BezLan16] is a short application of Lutwak’s
containment theorem ([Lu]) and might be of independent interest, we recall it
here. Clearly, $\lambda_{\rm
simplex}^{(d)}\leq\sup_{\mathcal{K}}\lambda(\mathcal{K})$. So, it is
sufficient to show that for every convex body $\mathbf{K}$ in
${\mathbb{E}}^{d}$ and every NS-family
$\mathcal{K}=\\{\mathbf{x}_{i}+\tau_{i}\mathbf{K}\ |\
\mathbf{x}_{i}\in{\mathbb{R}}^{d},\tau_{i}>0,i=1,2,\ldots,n\\}$ a translate of
$\lambda_{\rm
simplex}^{(d)}\left(\sum\limits_{i=1}^{n}\tau_{i}\right)\mathbf{K}$ covers
$\bigcup\mathcal{K}$. Now, according to Lutwak’s containment theorem ([Lu]) if
$\mathbf{K}_{1}$ and $\mathbf{K}_{2}$ are convex bodies in ${\mathbb{E}}^{d}$
such that every circumscribed simplex of $\mathbf{K}_{2}$ has a translate that
covers $\mathbf{K}_{1}$, then $\mathbf{K}_{2}$ has a translate that covers
$\mathbf{K}_{1}$. (Here a circumscribed simplex of $\mathbf{K}_{2}$ means a
$d$-simplex of ${\mathbb{E}}^{d}$ that contains $\mathbf{K}_{2}$ such that
each facet of the $d$-simplex meets $\mathbf{K}_{2}$.) Thus, if
$\Delta(\mathbf{K})$ is a circumscribed simplex of $\mathbf{K}$, then
$\lambda_{\rm
simplex}^{(d)}\left(\sum\limits_{i=1}^{n}\tau_{i}\right)\Delta(\mathbf{K})$ is
a circumscribed simplex of $\lambda_{\rm
simplex}^{(d)}\left(\sum\limits_{i=1}^{n}\tau_{i}\right)\mathbf{K}$ and
$\mathbf{x}_{i}+\tau_{i}\Delta(\mathbf{K})$ is a circumscribed simplex of
$\mathbf{x}_{i}+\tau_{i}\mathbf{K}$ for all $i=1,2,\ldots,n$. Furthermore,
$\\{\mathbf{x}_{i}+\tau_{i}\Delta(\mathbf{K})\ |\
\mathbf{x}_{i}\in{\mathbb{E}}^{d},\tau_{i}>0,i=1,2,\ldots,n\\}$ is an NS-
family and therefore $\lambda_{\rm
simplex}^{(d)}\left(\sum\limits_{i=1}^{n}\tau_{i}\right)\Delta(\mathbf{K})$
has a translate that covers
$\bigcup\\{\mathbf{x}_{i}+\tau_{i}\Delta(\mathbf{K})\ |\
\mathbf{x}_{i}\in{\mathbb{R}}^{d},\tau_{i}>0,i=1,2,\ldots,n\\}\supseteq\bigcup\mathcal{K}$,
which completes our proof via Lutwak’s containment theorem.

In connection with the question whether Theorem 2.4 can be extended to non-
symmetric convex bodies, it was proved in [BezLan16] that if
$\mathcal{K}=\\{\mathbf{x}_{i}+\tau_{i}\mathbf{K}:\mathbf{x}_{i}\in{\mathbb{E}}^{d},\tau_{i}>0,i=1,2,\ldots,n\\}$
is an arbitrary NS-family of positive homothetic copies of the convex body
$\mathbf{K}$ in ${\mathbb{E}}^{d}$, then $\lambda(\mathcal{K})\leq d$ holds
for all $n\geq 2$ and $d\geq 2$. This upper bound was recently improved by
Akopyan, Balitskiy, and Grigorev [ABG] as follows.

######  Theorem 2.6.

Let
$\mathcal{T}=\\{\mathbf{x}_{i}+\tau_{i}\mathbf{T}:\mathbf{x}_{i}\in{\mathbb{E}}^{d},\tau_{i}>0,i=1,2,\ldots,n\\}$
be an arbitrary family of positive homothetic copies of the $d$-dimensional
simplex $\mathbf{T}$ in ${\mathbb{E}}^{d}$ such that any hyperplane parallel
to a facet of $\mathbf{T}$ and intersecting ${\rm
conv}\left(\bigcup\mathcal{T}\right)$ intersects a member of $\mathcal{T}$.
Then it is possible to cover $\bigcup\mathcal{T}$ by a translate of
$\frac{d+1}{2}\left(\sum_{i=1}^{n}\tau_{i}\right)\mathbf{T}$. Moreover, the
factor $\frac{d+1}{2}$ cannot be improved.

Clearly, Lemma 2.5 and Theorem 2.6 imply

######  Corollary 2.7.

If
$\mathcal{K}=\\{\mathbf{x}_{i}+\tau_{i}\mathbf{K}:\mathbf{x}_{i}\in{\mathbb{E}}^{d},\tau_{i}>0,i=1,2,\ldots,n\\}$
is an arbitrary NS-family of positive homothetic copies of the convex body
$\mathbf{K}$ in ${\mathbb{E}}^{d}$, then
$\lambda(\mathcal{K})\leq\frac{d+1}{2}$ holds for all $n\geq 2$ and $d\geq 2$.

Hence, the following problem remains open.

######  Problem 1.

Find $\sup_{\mathcal{K}}\lambda(\mathcal{K})$ for any given $d\geq 2$, where
$\mathcal{K}$ runs over the NS-families of finitely many positive homothetic
copies of an arbitrary convex body $\mathbf{K}$ in ${\mathbb{E}}^{d}$. In
particular, is there an absolute constant $c>0$ such that
$\sup_{\mathcal{K}}\lambda(\mathcal{K})\leq c$ holds for all $d\geq 2$?

###  2.2. More on non-separable translative families of convex bodies

In this subsection we raise a number of isoperimetric-type questions for NS-
families of translates of convex bodies, and prove some partial results about
them.

######  Problem 2.

Let $\mathbf{K}$ be a convex body in ${\mathbb{E}}^{d}$. Find the maxima of
the quantities $\mathrm{vol}_{d}({\rm conv}\left(\bigcup\mathcal{F})\right)$,
$\mathrm{svol}_{d-1}\left({\rm conv}(\bigcup\mathcal{F})\right)$, $r\left({\rm
conv}(\bigcup\mathcal{F})\right)$ and $w\left({\rm
conv}(\bigcup\mathcal{F})\right)$, where $\mathcal{F}$ runs over all NS-
families of $m>1$ translates of $\mathbf{K}$, and $\mathrm{svol}_{d-1}(\cdot)$
and $w(\cdot)$ refer to the surface volume and the minimal width, respectively
of the corresponding convex body.

######  Remark 1.

It is easy to see that for any NS-family
$\mathcal{F}=\\{\mathbf{x}_{i}+\lambda_{i}\mathbf{K}:\lambda_{i}>0,i=1,2,\ldots,m\\}$
of positive homothetic copies of a convex body $\mathbf{K}$ in
${\mathbb{E}}^{d}$, we have $\operatorname{diam}\left({\rm
conv}(\mathcal{F})\right)\leq\left(\sum_{i=1}^{m}\lambda_{i}\right)\operatorname{diam}(\mathbf{K})$.
Indeed, recall that the diameter of a convex body coincides with its longest
orthogonal projection onto a line, and hence, the inequality follows from the
property that the orthogonal projection of an NS-family of convex bodies onto
any line is a closed segment. In this inequality we have equality, if and only
if, $\mathbf{K}$ has a diameter $[\mathbf{p},\mathbf{q}]$ such that for any
$2\leq i\leq m$,
$\mathbf{x}_{i-1}+\lambda_{i-1}\mathbf{q}=\mathbf{x}_{i}+\lambda_{i}\mathbf{p}$.

We start the investigation with the following variant of Kirchberger’s theorem
([Sch14]).

######  Theorem 2.8.

Let $\mathcal{F}=\\{\mathbf{K}_{i}:i=1,2,\ldots,m\\}$ be a family of convex
bodies in ${\mathbb{E}}^{d}$. Let $\mathcal{F}_{1}$ and $\mathcal{F}_{2}$
satisfy $\mathcal{F}_{1}\cap\mathcal{F}_{2}=\emptyset$,
$\mathcal{F}_{1}\cup\mathcal{F}_{2}=\mathcal{F}$. Then there is a hyperplane
(strictly) separating $\bigcup\mathcal{F}_{1}$ from $\bigcup\mathcal{F}_{2}$
if and only if for any subfamily $\mathcal{G}$ of $\mathcal{F}$ of cardinality
at most $d+2$, $\bigcup(\mathcal{F}_{1}\cap\mathcal{G})$ can be (strictly)
separated by a hyperplane from $\bigcup(\mathcal{F}_{2}\cap\mathcal{G})$.

###### Proof.

For any $i=1,2,\ldots,m$ with $\mathbf{K}_{i}\in\mathcal{F}_{1}$, let
$\hat{\mathbf{K}}_{i}\subset\mathbb{E}^{d+1}$ be the set of points
$(\mathbf{y},\alpha)$ with $\mathbf{y}\in{\mathbb{E}}^{d}$ and
$\alpha\in{\mathbb{R}}$ such that $\mathbf{K}_{i}$ is contained in the closed
half space
$\\{\mathbf{x}\in{\mathbb{R}}^{n}:\langle\mathbf{x},\mathbf{y}\rangle\leq\alpha\\}$.
Then $\hat{\mathbf{K}}_{i}$ is convex. Indeed, let
$(\mathbf{y}_{1},\alpha_{1}),(\mathbf{y}_{2},\alpha_{2})\in\hat{\mathbf{K}}_{i}$,
and $0\leq t\leq 1$. Consider any $\mathbf{x}\in\mathbf{K}$. Then for $s=1,2$,
$\langle\mathbf{x},\mathbf{y}_{s}\rangle\leq\alpha_{s}$, and

$\langle\mathbf{x},t\mathbf{y}_{1}+(1-t)\mathbf{y}_{2}\rangle=t\langle\mathbf{x},\mathbf{y}_{1}\rangle+(1-t)\langle\mathbf{x},\mathbf{y}_{2}\rangle\leq
t\alpha_{1}+(1-t)\alpha_{2},$

implying that
$t(\mathbf{y}_{1},\alpha_{1})+(1-t)(\mathbf{y}_{2},\alpha_{2})\in\hat{\mathbf{K}}_{i}$.
It follows similarly that for any $i=1,2,\ldots,m$ with
$\mathbf{K}_{i}\in\mathcal{F}_{2}$, the set
$\hat{\mathbf{K}}_{i}\subset\mathbb{E}^{d+1}$ of points $(\mathbf{y},\alpha)$
with $\mathbf{y}\in{\mathbb{E}}^{d}$ and $\alpha\in{\mathbb{R}}$ such that
$\mathbf{K}_{i}$ is contained in the closed half space
$\\{\mathbf{x}\in{\mathbb{E}}^{d}:\langle\mathbf{x},\mathbf{y}\rangle\geq\alpha\\}$,
is convex. Thus, the assertion for nonstrict separation follows from Helly’s
theorem, and a straightforward modification shows it for strict separation. ∎

######  Corollary 2.9.

Let $\mathcal{F}=\\{\mathbf{K}_{i}:i=1,2,\ldots,m\\}$ be a family of convex
bodies in ${\mathbb{E}}^{d}$. Then $\mathcal{F}$ is non-separable if and only
if the following holds: for any partition of $\mathcal{F}$ into subfamilies
$\mathcal{F}_{1}$ and $\mathcal{F}_{2}$, there is a subfamily $\mathcal{G}$ of
$\mathcal{F}$ of cardinality at most $d+2$ such that
$\bigcup(\mathcal{F}_{1}\cap\mathcal{G})$ is not strictly separated from
$\bigcup(\mathcal{F}_{2}\cap\mathcal{G})$ by a hyperplane.

Our next theorem answers Problem 2 for the case of three Euclidean unit disks.

######  Theorem 2.10.

Let $\mathcal{F}=\\{\mathbf{B}[\mathbf{x}_{i},1]:i=1,2,3\\}$ be an NS-family
of three unit disks in $\mathbb{E}^{2}$. Let $\mathbf{T}={\rm
conv}\\{\mathbf{x}_{1},\mathbf{x}_{2},\mathbf{x}_{3}\\}$ and
$\mathbf{C}=\mathbf{T}+\mathbf{B}[\mathbf{o},1]$. Then

  1. (a)

$\operatorname{area}(\mathbf{C})\leq\pi+4+3\sqrt{3}$, with equality if and
only if $\mathbf{T}$ is an isosceles triangle such that its legs are of length
$2$, and the angle between them is $\frac{2\pi}{3}$.

  2. (b)

$\operatorname{per}(\mathbf{C})\leq 2\pi+8$, with equality if an only if
$\mathbf{T}$ is a segment of length $4$.

  3. (c)

$r(\mathbf{C})\leq\frac{16}{3}$, with equality if and only if $\mathbf{T}$ is
a regular triangle whose heights are of length $2$.

  4. (d)

$w(\mathbf{C})\leq 4$, with equality if and only if $\mathbf{T}$ is a regular
triangle whose heights are of length $2$.

###### Proof.

First, we prove (a). By Steiner’s formula, we have

$\operatorname{area}(\mathbf{C})=\pi+\operatorname{area}(\mathbf{T})+\operatorname{per}(\mathbf{T}).$

First, consider the case that $\mathbf{T}$ has a non-acute angle $\gamma$.
Then the property that $\mathcal{F}$ is non-separable is equivalent to the
property that the two sides of $\mathbf{T}$ enclosing this angle have lengths
at most $2$. Thus, we may assume that these lenghts are equal to $2$. In this
case

$\operatorname{area}(\mathbf{C})=\pi+4+4\sin\frac{\gamma}{2}+2\sin\gamma.$

We denote the expression on the right-hand side by $A_{o}(\gamma)$. Since the
sine function is strictly concave on $(0,\pi)$, it follows that
$A_{o}(\gamma)$ is strictly concave on $[\pi/2,\pi)$, and it has a unique
maximum. Thus, by differentiating, we obtain that $A_{0}(\gamma)$ is maximal
if and only if $\gamma=\gamma=\frac{2\pi}{3}$.

Next, consider the case that $\mathbf{T}$ has no obtuse angles. In this case
the property that $\mathcal{F}$ is non-separable is equivalent to the property
that no height of $\mathbf{T}$ has length greater than $2$. For $i=1,2,3$, let
the side of $\mathbf{T}$ opposite of $\mathbf{x}_{i}$ be denoted by $S_{i}$,
and its length by $a_{i}$. Assume that $a_{1}\leq a_{2}\leq a_{3}$. Let
$h_{i}$ denote the length of the height of $\mathbf{T}$ through
$\mathbf{x}_{i}$. Since $a_{i}h_{i}=2\operatorname{area}(T)$ for all values of
$i$, it follows that $h_{3}\leq h_{2}\leq h_{1}$, and we may assume that
$h_{1}=2$. Note that if we move $\mathbf{x}_{1}$ on the line parallel to
$[\mathbf{x}_{2},\mathbf{x}_{3}]$ and away from the bisector of the segment,
then $h_{1}$ as well as $\operatorname{area}(\mathbf{T})$ do not change, but
$\operatorname{per}(\mathbf{T})$ increases. Thus, if
$\operatorname{area}(\mathbf{C})$ is maximal, then $h_{2}=h_{1}=2$, or
$\mathbf{T}$ has a right angle at $\mathbf{x}_{3}$. By the argument in the
previous paragraph, it is sufficient to consider the case that
$h_{1}=h_{2}=2$.

Let $\frac{\pi}{3}\leq\gamma\leq\frac{\pi}{2}$ denote the angle of
$\mathbf{T}$ at $\mathbf{x}_{3}$. Then $a_{1}=a_{2}=\frac{2}{\sin\gamma}$, and

$\operatorname{area}(\mathbf{C})=\pi+\frac{4}{\sin\gamma}+\frac{4\sin\frac{\gamma}{2}}{\sin\gamma}+\frac{2}{\sin\gamma}=\pi+\frac{6}{\sin\gamma}+\frac{2}{\cos\frac{\gamma}{2}}.$

We denote the expression on the right-hand side by $A_{a}(\gamma)$. By
differentiation, we obtain that if $A_{a}(\gamma)$ is maximal, then
$\gamma=\frac{\pi}{3}$ or $\gamma=\frac{\pi}{2}$. Thus, (a) follows from the
inequalities
$A_{a}\left(\frac{\pi}{3}\right)<A_{a}\left(\frac{\pi}{2}\right)<A_{o}\left(\frac{2\pi}{3}\right)$.

Next, we prove (b). Observe that

$\operatorname{per}(\mathbf{C})=2\pi+\operatorname{per}(\mathbf{T}).$

Following the method of the proof of (a), we have that if $\mathbf{T}$ is not
acute, then, by the triangle inequality, $\operatorname{per}(\mathbf{T})\leq
8$. Similarly, if $\mathbf{T}$ has no obtuse angles, than the maximality of
$\operatorname{per}(\mathbf{C})$ yields that the two shortest sides of $T$
have equal lengths, and the length of the corresponding heights is $2$. In
this case, denoting by $\gamma$ the angle enclosed by the two legs,

$\operatorname{per}(\mathbf{T})=\frac{4}{\sin\gamma}+\frac{4\sin\frac{\gamma}{2}}{\sin\gamma}.$

By differentiating, it follows that the right-hand side is maximal if
$\gamma=\frac{\pi}{2}$, and thus, if $\mathbf{T}$ is not obtuse, then
$\operatorname{per}(\mathbf{T})\leq 4+2\sqrt{2}<8$, implying (b).

To prove (c), we observe that $w(\mathbf{C})=w(\mathbf{T})+2$ and that
$w(\mathbf{T})$ coincides with the length of the shortest height of
$\mathbf{T}$. To prove (d), we note that $r(\mathbf{C})=r(\mathbf{T})+2$, and
apply a case analysis like in the proofs of (a) and (b). ∎

To present our next result, we need a definition. We note that the same
concept for a family of segments was defined by Siegel in [Siegel].

######  Definition 2.

Let $\mathcal{F}=\\{\mathbf{K}_{i}:i=1,2,\ldots,m\\}$ be a family of $m$
convex bodies in ${\mathbb{E}}^{d}$. If we can relabel the indices of the
elements such that for any $i=2,3,\ldots,m$, $\mathbf{x}_{i}+\mathbf{K}$
intersects $\bigcup_{j=1}^{i-1}(\mathbf{x}_{j}+\mathbf{K})$, we say that
$\mathcal{F}$ is _a successively non-separable family_ , in short, an _SNS-
family_.

######  Remark 2.

Every SNS-family is non-separable, but there are NS-families that are not
successively non-separable.

###### Proof.

Let $\mathcal{F}=\\{\mathbf{K}_{i}:i=1,2,\ldots,m\\}$ be an SNS-family, where
the indices are chosen to satisfy the condition in Definition 2. For
contradiction, suppose that there is a hyperplane $H$ strictly separating some
of the elements of $\mathcal{F}$ from the rest of the elements. Let $i>1$
denote the smallest index such that $\mathbf{K}_{i}$ is strictly separated
from $\mathbf{K}_{1}$. Then $H$ strictly separates $\mathbf{K}_{i}$ from
$\bigcup_{j=1}^{i-1}\mathbf{K}_{j}$, and thus, also from ${\rm
conv}\bigcup_{j=1}^{i-1}\mathbf{K}_{j}$, contradicting our assumption that
$\mathcal{F}$ is an SNS-family. As an example of an NS-family that is not
sucessively non-separable, we refer the reader to the configuration in Figure
1. ∎

######  Theorem 2.11.

Let $\mathcal{F}$ be an SNS-family of $m$ unit disks in $\mathbb{E}^{2}$. Then

$\operatorname{per}\left({\rm conv}\bigcup\mathcal{F}\right)\leq 2\pi+2m-2,$

with equality, if and only if, the centerpoints of the elements of
$\mathcal{F}$ are all collinear and the distance between any two consecutive
centerpoints on this line is $2$.

Note that Theorem 2.11 is trivial for $m=1$. Thus, it readily follows from
Lemma 2.12 by applying induction on $m$.

######  Lemma 2.12.

Let $\mathbf{K}$ be the convex hull of finitely many unit disks. Let
$\mathbf{B}=\mathbf{B}[\mathbf{x},1]$ satisfy
$\mathbf{B}\cap\mathbf{K}\neq\emptyset$. Then

$\operatorname{per}({\rm
conv}(\mathbf{K}\cup\mathbf{B}))\leq\operatorname{per}(\mathbf{K})+4,$

with equality, if and only if, $\mathbf{B}$ touches $\mathbf{K}$, and the
centers of all unit disks generating $\mathbf{K}$, and that of $\mathbf{B}$,
are collinear.

In the proof we denote the arclength of a curve $\Gamma$ by $\ell(\Gamma)$.

###### Proof of Lemma 2.12.

Without loss of generality, assume that $\mathbf{B}$ touches $\mathbf{K}$ at
some point $\mathbf{p}$. Then ${\rm bd}({\rm conv}(K\cup B))$ consists of an
arc in ${\rm bd}(K)$, two segments $[\mathbf{p}_{i},\mathbf{q}_{i}]$, where
$i=1,2$, connecting a point $\mathbf{p}_{i}\in{\rm bd}(\mathbf{K})$ and a
point $\mathbf{q}_{i}\in{\rm bd}(\mathbf{B})$, and a unit circle arc in ${\rm
bd}(\mathbf{B})$ connecting $\mathbf{q}_{1}$ and $\mathbf{q}_{2}$.

Let $\mathbf{q}$ denote the point of $\mathbf{B}$ antipodal to $\mathbf{p}$,
and note that $\mathbf{q}\in{\rm bd}(\mathbf{K}\cup\mathbf{B})$. For $i=1,2$,
let $\Gamma_{i}$ denote the arc of ${\rm bd}(\mathbf{K})$ inside ${\rm
conv}(\mathbf{K}\cup\mathbf{B})$ from $\mathbf{p}$ to $\mathbf{p}_{i}$, and
let $\Delta_{i}$ denote the union of $[\mathbf{p}_{i},\mathbf{q}_{i}]$ and the
shorter arc $\Theta_{i}$ of ${\rm bd}(\mathbf{B})$ from $\mathbf{q}$ to
$\mathbf{q}_{i}$. We need to show that for $i=1,2$,
$\ell(\Delta_{i})\leq\ell(\Gamma_{i})+2$.

To do it, observe that ${\rm bd}(\mathbf{K})$ consists of segments and unit
circle arcs. Furthermore, the total length of the unit circle arcs in an arc
in ${\rm bd}(K)$ is equal to the total turning angle of the arc as we move
from one endpoint to the other one. Since at $\mathbf{p}_{i}$, $\mathbf{K}$
and ${\rm conv}(\mathbf{K}\cup\mathbf{B})$ have the same tangent line and the
tangent line of $\mathbf{K}$ at $\mathbf{p}$ and that of ${\rm
conv}(\mathbf{K}\cup\mathbf{B})$ at $\mathbf{q}$ are parallel, the unit circle
arcs in $\Gamma_{i}$ can be obtained by subdividing $\Theta_{i}$ into finitely
many arcs, and translating each part by a vector. Thus, we can translate the
segments and unit circle arcs in $\Gamma_{i}$ to obtain the union of the
translate $\Theta_{i}^{\prime}$ of $\Theta_{i}$ by the vector
$\mathbf{p}-\mathbf{q}$, and a polygonal curve connecting
$\mathbf{q}_{i}^{\prime}$ to $\mathbf{p}_{i}$, where $\mathbf{q}_{i}^{\prime}$
is the translate of $\mathbf{q}_{i}$ by $\mathbf{p}-\mathbf{q}$. Thus, by the
triangle inequality, we have

$\ell(\Delta_{i})=\ell(\Theta_{i})+|\mathbf{q}_{i}-\mathbf{p}_{i}|\leq\ell(\Theta_{i}^{\prime})+|\mathbf{p}_{i}-\mathbf{q}_{i}^{\prime}|+|\mathbf{q}_{i}^{\prime}-\mathbf{q}_{i}|\leq\Gamma_{i}+2.$

Equality here implies that
$\mathbf{q}_{i},\mathbf{q}_{i}^{\prime},\mathbf{p}_{i}$ are collinear, and
$\Gamma_{i}$ is the union of $\Theta_{i}^{\prime}$ and
$[\mathbf{p}_{i},\mathbf{q}_{i}^{\prime}]$, which clearly yields that
$\mathbf{q}_{i}^{\prime}=\mathbf{p}_{i}$ and the length of $\Theta_{i}$ is
$\frac{\pi}{2}$. ∎

######  Remark 3.

A straightforward modification of the proof of Lemma 2.12 yields that for any
SNS-family $\mathcal{F}$ of $k$ translates of an $\mathbf{o}$-symmetric convex
domain $\mathbf{M}$, we have $\operatorname{per}_{M}({\rm
conv}(\mathcal{F}))\leq 2k-2+\operatorname{per}_{M}(\mathbf{M})$, where
$\operatorname{per}_{M}(\cdot)$ denotes perimeter measured in the norm of
$\mathbf{M}$.

######  Conjecture 1.

For any NS-family $\mathcal{F}$ of $k$ translates of an $\mathbf{o}$-symmetric
convex domain $\mathbf{M}$, we have $\operatorname{per}_{M}({\rm
conv}(\mathcal{F}))\leq 2k-2+\operatorname{per}_{M}(\mathbf{M})$.

Before Corollary 2.13, recall that the mean width
$\operatorname{mw}(\mathbf{K})$ of the plane convex body $\mathbf{K}$
satisfies the equality
$\operatorname{per}(\mathbf{K})=\pi\operatorname{mw}(\mathbf{K})$ (see e.g.
[Gardner]).

######  Corollary 2.13.

For any SNS-family $\mathcal{F}$ of $m$ unit disks,

$\operatorname{mw}\left({\rm conv}\bigcup\mathcal{F}\right)\leq
2+\frac{2m}{\pi},$

with equality, if and only if, the centerpoints of the elements of
$\mathcal{F}$ are all collinear and the distance between any two consecutive
centerpoints on this line is $2$.

######  Corollary 2.14.

For any SNS-family $\mathcal{F}$ of $m$ unit balls in ${\mathbb{E}}^{d}$, we
have

$\operatorname{mw}\left(\bigcup\mathcal{F}\right)\leq 2+\frac{2m}{\pi}.$

###### Proof.

By formulas (A.46) and (A.50) of [Gardner], for any convex body $\mathbf{K}$
in ${\mathbb{E}}^{d}$,

$\operatorname{mw}(\mathbf{K})=\int_{\mathcal{G}(d,2)}\operatorname{mw}(\mathbf{K}|S)\,dS,$

where $\mathcal{G}(d,k)$ denotes the Grassmannian of the $k$-dimensional
linear subspaces of in ${\mathbb{E}}^{d}$, $\mathbf{K}|S$ denotes the
orthogonal projection of $\mathbf{K}$ onto the linear subspace $S$, and the
integration is with respect to the unique uniform probability measure of
$\mathcal{G}(d,2)$. Thus, Corollary 2.14 is an immediate consequence of
Theorem 2.11. ∎

###  2.3. Non-separable arrangements in spherical spaces

In order to state the main results of this section we need to recall some
definitions. A closed cap, in short a cap, of spherical radius $\alpha$, for
$0\leq\alpha\leq\pi$, is the set of points with spherical distance at most
$\alpha$ from a given point in ${\mathbb{S}}^{d-1}\subset{\mathbb{E}}^{d}$. A
great sphere of ${\mathbb{S}}^{d-1}$ is an intersection of
${\mathbb{S}}^{d-1}$ with a hyperplane of ${\mathbb{E}}^{d}$ passing through
the origin $\mathbf{o}\in{\mathbb{E}}^{d}$. Following the terminology of
Polyanskii [Po21], we say that a great sphere avoids a collection of caps in
${\mathbb{S}}^{d-1}$ if it does not intersect any cap of the collection.
Finally, we say that a finite collection of caps is non-separable if it does
not have a great sphere that avoids the caps such that on both sides of it
there is at least one cap. Based on these concepts Polyanskii [Po21] very
recently proved the following extension of Theorem 2.1 to spherical spaces.

######  Theorem 2.15.

Let $\mathcal{F}$ be a non-separable family of caps of spherical radii
$\alpha_{1},\dots,\alpha_{n}$ in ${\mathbb{S}}^{d-1}$, $d\geq 2$. If
$\alpha_{1}+\dots+\alpha_{n}<\frac{\pi}{2}$, then $\mathcal{F}$ can be covered
by a cap of radius $\alpha_{1}+\dots+\alpha_{n}$ in ${\mathbb{S}}^{d-1}$.

It is shown in [Po21] that Theorem 2.15 is equivalent to Theorem 2.16 below.
Polyanskii’s proof of Theorem 2.16 uses ideas from [Bal], [Ball], and [Bang].
Recall that a closed zone, in short a zone, of width $2\alpha$ with
$0\leq\alpha\leq\frac{\pi}{2}$ in ${\mathbb{S}}^{d-1}$ is the set of points
with spherical distance at most $\alpha$ from a given great sphere in
${\mathbb{S}}^{d-1}$. As a spherically convex body is the intersection of
${\mathbb{S}}^{d-1}$ with a $d$-dimensional closed convex cone of
${\mathbb{E}}^{d}$ different from ${\mathbb{E}}^{d}$, therefore a zone of
width $2\alpha$ for $0<\alpha<\frac{\pi}{2}$ is not a spherically convex body.

######  Theorem 2.16.

Let
$\mathbf{Z}_{1}\subset{\mathbb{S}}^{d-1},\dots,\mathbf{Z}_{n}\subset{\mathbb{S}}^{d-1}$,
$d\geq 2$ be zones of width $2\alpha_{1},\dots,2\alpha_{n}$, respectively,
such that $\alpha_{1}+\dots+\alpha_{n}<\frac{\pi}{2}$. If
${\mathbb{S}}^{d-1}\setminus\left(\bigcup_{i=1}^{n}\mathbf{Z}_{i}\right)$
possesses at most one pair of antipodal open connected components, then
$\bigcup_{i=1}^{n}\mathbf{Z}_{i}$ can be covered by a zone of width
$2\alpha_{1}+\dots+2\alpha_{n}$.

According to [Po21], Maxim Didid suggested to investigate the following more
general problem, which he phrased as a conjecture. In what follows, the
inradius of a spherically convex body is the spherical radius of the largest
cap contained in it.

######  Conjecture 2.

Let
$\mathbf{Z}_{1}\subset{\mathbb{S}}^{d-1},\dots,\mathbf{Z}_{n}\subset{\mathbb{S}}^{d-1}$,
$d>2$ be zones of width $2\beta_{1},\dots,2\beta_{n}$, respectively. If
${\mathbb{S}}^{d-1}\setminus\left(\bigcup_{i=1}^{n}\mathbf{Z}_{i}\right)$
consists of $2m$ spherically convex open connected components with inradii
$\gamma_{1},\dots,\gamma_{2m}$, respectively, then $2\beta_{1}+\dots
2\beta_{n}+\gamma_{1}+\dots+\gamma_{2m}\geq\pi$.

######  Remark 4.

The first named author and Schneider [BezSch] proved that if a cap of
spherical radius $\alpha\geq\frac{\pi}{2}$ is covered by a finite family of
spherically convex bodies in ${\mathbb{S}}^{d-1}$, $d>2$, then the sum of the
inradii of the spherically convex bodies in the family is at least $\alpha$.
Clearly, this theorem implies Conjecture 2 for the case when
$\beta_{1}=\dots=\beta_{n}=0$, i.e., $\mathbf{Z}_{1},\dots,\mathbf{Z}_{n}$ are
great spheres of ${\mathbb{S}}^{d-1}$.

In connection with Remark 4 we call the reader’s attention to Problem 7.3.5 of
[Be13], which is still open and so, we include it here.

######  Problem 3.

Prove or disprove that if a cap of spherical radius $0<\alpha<\frac{\pi}{2}$
is covered by a finite family of spherically convex bodies in
${\mathbb{S}}^{d-1}$, $d>2$, then the sum of the inradii of the spherically
convex bodies in the family is at least $\alpha$.

######  Remark 5.

Jiang and Polyanskii [JiPo] proved that if a finite family of zones covers
${\mathbb{S}}^{d-1}$, then the sum of the widths of the zones in the family is
at least $\pi$. Ortega-Moreno [OM] has found another proof, which was
simplified by Zhao [Zh] (see also [GKP]). Clearly, this theorem implies
Conjecture 2 for the case when $\gamma_{1}=\dots=\gamma_{2m}=0$, i.e.,
${\mathbb{S}}^{d-1}=\bigcup_{i=1}^{n}\mathbf{Z}_{i}$.

##  3\. Dense totally separable packings

###  3.1. Finding the densest totally separable (finite or infinite)
translative packings in the Euclidean plane

The concept of totally separable packings was introduced by Fejes Tóth and
Fejes Tóth [FeFe] as follows.

######  Definition 3.

A packing $\mathcal{F}$ of convex domains in ${\mathbb{E}}^{2}$ is called a
_totally separable packing_ , in short, a _TS-packing_ , if any two members of
$\mathcal{F}$ can be separated by a line which is disjoint from the interiors
of all members of $\mathcal{F}$.

We shall use the following notation.

######  Definition 4.

Let $\mathbf{K}$ be a convex domain in ${\mathbb{E}}^{2}$. Then let
$\square^{*}(\mathbf{K})$ (resp., $\square(\mathbf{K})$) denote a minimal area
circumscribed quadrilateral (resp., parallelogram) of $\mathbf{K}$.

Fejes Tóth and Fejes Tóth [FeFe] put forward the problem of finding the
largest density of TS-packings by congruent copies of a given convex domain in
${\mathbb{E}}^{2}$. They have solved this problem for centrally symmetric
convex domains as follows.

######  Theorem 3.1.

Let $\mathbf{K}$ be a convex domain in ${\mathbb{E}}^{2}$ and let
$\mathbf{Q}\subset{\mathbb{E}}^{2}$ be a convex quadrilateral that contains
$n>1$ congruent copies of $\mathbf{K}$ forming a TS-packing in $\mathbf{Q}$.
Then ${\rm area}(\mathbf{Q})\geq n\cdot{\rm area}(\square^{*}(\mathbf{K}))$.

According to a theorem of Dowker [Dow] if $\mathbf{K}\subset{\mathbb{E}}^{2}$
is a centrally symmetric convex domain, then among the least area convex
quadrilaterals containing $\mathbf{K}$ there is a parallelogram. Clearly, this
observation and Theorem 3.1 imply

######  Corollary 3.2.

Let $\mathbf{K}$ be a centrally symmetric convex domain and let $\mathcal{P}$
be an arbitrary TS-packing by congruent copies of $\mathbf{K}$. Then for the
density $\delta(\mathcal{P})$ of $\mathcal{P}$ (i.e., for the fraction
$\delta(\mathcal{P})$ of ${\mathbb{E}}^{2}$ covered by the members of
$\mathcal{P}$) we have that $\delta(\mathcal{P})\leq\frac{{\rm
area}(\mathbf{K})}{{\rm area}(\square(\mathbf{K}))}$. Here, equality is
attained for the lattice TS-packing of $\mathbf{K}$ with fundamental
parallelogram $\square(\mathbf{K})$.

The line of research started in [FeFe] has been continued by the authors in
[BeLa20]. On the one hand, the following close relative of Corollary 3.2 was
proved in [BeLa20].

######  Theorem 3.3.

If $\delta_{sep}(\mathbf{K})$ denotes the largest (upper) density of TS-
packings by translates of the convex domain $\mathbf{K}$ in
${\mathbb{E}}^{2}$, then

(2) $\delta_{sep}(\mathbf{K})=\frac{{\rm area}(\mathbf{K})}{{\rm
area}(\square(\mathbf{K}))}.$

######  Remark 6.

It is worth pointing out that by (2) of Theorem 3.3, the densest TS-packing by
translates of a convex domain is attained by a lattice packing.

On the other hand, the following finite TS-packing analogue of Theorem 3.3 was
proved by the authors in [BeLa20].

######  Theorem 3.4.

Let $\mathcal{F}=\\{\mathbf{c}_{i}+\mathbf{K}:i=1,2,\ldots,n\\}$ be a TS-
packing by $n$ translates of the convex domain $\mathbf{K}$ in
${\mathbb{E}}^{2}$. Let $\mathbf{C}={\rm
conv}\left(\\{\mathbf{c}_{1},\mathbf{c}_{2},\ldots,\mathbf{c}_{n}\\}\right)$.

  1. (3.4.1)

Then we have

${\rm area}\left({\rm
conv}\left(\bigcup_{i=1}^{n}(\mathbf{c}_{i}+\mathbf{K})\right)\right)={\rm
area}(\mathbf{C}+\mathbf{K})\geq\frac{2}{3}(n-1){\rm
area}\left(\square(\mathbf{K})\right)+{\rm area}(\mathbf{K})+\frac{1}{3}{\rm
area}(\mathbf{C}).$

  2. (3.4.2)

If $\mathbf{K}$ or $\mathbf{C}$ is centrally symmetric, then

${\rm area}(\mathbf{C}+\mathbf{K})\geq(n-1){\rm
area}\left(\square(\mathbf{K})\right)+{\rm area}(\mathbf{K}).$

######  Remark 7.

We note that equality is attained in $(\ref{thm:areaformula}.1)$ of Theorem
3.4 for the following TS-packings by translates of a triangle (cf. Figure 3).
Let $\mathbf{K}$ be a triangle, with the origin $\mathbf{o}$ at a vertex, and
$\mathbf{u}$ and $\mathbf{v}$ being the position vectors of the other two
vertices, and let $\mathbf{T}=m\mathbf{K}$, where $m>1$ is an integer. Let
$\mathcal{F}$ be the family consisting of the elements of the lattice packing
$\\{i\mathbf{u}+j\mathbf{v}+\mathbf{K}:i,j,\in\mathbb{Z}\\}$ contained in
$\mathbf{T}$. Then $\mathcal{F}$ is a TS-packing by $n=\frac{m(m+1)}{2}$
translates of $\mathbf{K}$ with ${\rm
conv}\left(\bigcup\mathcal{F}\right)=\mathbf{T}=\mathbf{C}+\mathbf{K}$, where
$\mathbf{C}=(m-1)\mathbf{K}$. Thus, ${\rm area}(\mathbf{T})=m^{2}{\rm
area}(\mathbf{K})=\left[\frac{2}{3}m(m+1)-\frac{1}{3}+\frac{1}{3}(m-1)^{2}\right]{\rm
area}(\mathbf{K})=\frac{4}{3}(n-1){\rm area}(\mathbf{K})+{\rm
area}(\mathbf{K})+\frac{1}{3}{\rm area}(\mathbf{C})=$ $\frac{2}{3}(n-1){\rm
area}(\square(\mathbf{K}))+{\rm area}(\mathbf{K})+\frac{1}{3}{\rm
area}(\mathbf{C})$.

Figure 3. An example for equality in (3.4.1).

######  Remark 8.

In (3.4.2) of Theorem 3.4 equality can be attained in a variety of ways shown
in Figure 4 for both cases namely, when $\mathbf{C}$ is centrally symmetric
(and $\mathbf{K}$ is not centrally symmetric such as a triangle) and when
$\mathbf{K}$ is centrally symmetric (such as a circular disk) without any
assumption on the symmetry of $\mathbf{C}$.

Figure 4. TS-packings by translates of a triangle and a unit disk for which
equality is attained in (3.4.2) in Theorem 3.4.

The following is an easy corollary of (3.4.2) of Theorem 3.4, which is also a
close relative of Theorem 3.1.

######  Corollary 3.5.

Let $\mathcal{F}=\\{\mathbf{c}_{i}+\mathbf{K}:i=1,2,\ldots,n\\}$ be a TS-
packing by $n$ translates of the convex domain $\mathbf{K}$ in
${\mathbb{E}}^{2}$. Let $\mathbf{C}={\rm
conv}\left(\\{\mathbf{c}_{1},\mathbf{c}_{2},\ldots,\mathbf{c}_{n}\\}\right)$.
Furthermore, let $\mathbf{Q}\subset{\mathbb{E}}^{2}$ be a convex quadrilateral
that contains $\mathbf{C}+\mathbf{K}$. If $\mathbf{K}$ or $\mathbf{C}$ be
centrally symmetric in ${\mathbb{E}}^{2}$, then ${\rm
area}(\mathbf{Q})\geq(n-1){\rm area}\left(\square(\mathbf{K})\right)+{\rm
area}\left(\square^{*}(\mathbf{K})\right)$.

The proofs of Theorems 3.3 and 3.4 in [BeLa20] are based on a translative TS-
packing analogue of Oler’s inequality [Oler]. As it might be of independent
interest, we state it as follows. First, we recall the necessary definitions.

If $\mathbf{K}$ is an $\mathbf{o}$-symmetric convex domain in
${\mathbb{E}}^{2}$, then let $|\cdot|_{\mathbf{K}}$ denote the norm generated
by $\mathbf{K}$, i.e., let
$|\mathbf{x}|_{\mathbf{K}}=\min\\{\lambda:\mathbf{x}\in\lambda\mathbf{K}\\}$
for any $\mathbf{x}\in{\mathbb{E}}^{2}$. The distance between the points
$\mathbf{p}$ and $\mathbf{q}$ of ${\mathbb{E}}^{2}$ measured in the norm
$|\cdot|_{\mathbf{K}}$ is denoted by $|\mathbf{p}-\mathbf{q}|_{K}$. For the
sake of simplicity, the Euclidean distance between the points $\mathbf{p}$ and
$\mathbf{q}$ of ${\mathbb{E}}^{2}$ is denoted by $|\mathbf{p}-\mathbf{q}|$.

If $P=\bigcup_{i=1}^{n}[\mathbf{x}_{i-1},\mathbf{x}_{i}]$ is a polygonal curve
in ${\mathbb{E}}^{2}$ with $[\mathbf{x}_{i-1},\mathbf{x}_{i}]$ standing for
the closed line segment connecting $\mathbf{x}_{i-1}$ and $\mathbf{x}_{i}$,
and $\mathbf{K}$ is an $\mathbf{o}$-symmetric convex domain in
${\mathbb{E}}^{2}$, then the Minkowski length of $P$ is defined as
$M_{\mathbf{K}}(P)=\sum_{i=1}^{n}|\mathbf{x}_{i}-\mathbf{x}_{i-1}|_{\mathbf{K}}$.
Based on this and using approximation by polygonal curves one can define the
Minkowski length $M_{\mathbf{K}}(G)$ of any rectifiable curve
$G\subseteq{\mathbb{E}}^{2}$ in the norm $|\cdot|_{\mathbf{K}}$.

######  Definition 5.

A closed polygonal curve
$P=\bigcup_{i=1}^{m}[\mathbf{x}_{i-1},\mathbf{x}_{i}]$, where
$\mathbf{x}_{0}=\mathbf{x}_{m}$, is called _permissible_ if there is a
sequence of simple closed polygonal curves
$P^{n}=\bigcup_{i=1}^{m}[\mathbf{x}^{n}_{i-1},\mathbf{x}^{n}_{i}]$, where
$\mathbf{x}^{n}_{0}=\mathbf{x}^{n}_{m}$, satisfying
$\mathbf{x}^{n}_{i}\to\mathbf{x}_{i}$ for every value of $i$. The interior
${\rm int}(P)$ is defined as $\lim_{n\to\infty}{\rm int}(P^{n})$.

######  Remark 9.

By the properties of limits, if
$P=\bigcup_{i=1}^{m}[\mathbf{x}_{i-1},\mathbf{x}_{i}]$ is permissible and
$P^{n}$ and $Q^{n}$ are sequences of simple closed polygonal curves with
$\lim_{n\to\infty}P^{n}=\lim_{n\to\infty}Q^{n}=P$, then $\lim_{n\to\infty}{\rm
int}(P^{n})=\lim_{n\to\infty}{\rm int}(Q^{n})$, i.e., the interior of a
permissible curve is well defined.

One of the main results of [BeLa20] is the following translative TS-packing
analogue of Oler’s inequality [Oler].

######  Theorem 3.6.

Let $\mathbf{K}$ be an $\mathbf{o}$-symmetric convex domain in
${\mathbb{E}}^{2}$. Let

$\mathcal{F}=\\{\mathbf{x}_{i}+\mathbf{K}:i=1,2,\ldots,n\\}$

be a TS-packing by $n$ translates of $\mathbf{K}$ in ${\mathbb{E}}^{2}$, and
set $X=\\{\mathbf{x}_{1},\mathbf{x}_{2},\ldots,\mathbf{x}_{n}\\}$.
Furthermore, let $\Pi$ be a permissible closed polygonal curve with the
following properties:

  1. (1)

the vertices of $\Pi$ are points of $X$

and

  2. (2)

$X\subseteq\Pi^{*}$ with $\Pi^{*}=\Pi\cup{\rm int}(\Pi)$.

Then

(3) $\frac{{\rm area}(\Pi^{*})}{{\rm
area}\left(\square(\mathbf{K})\right)}+\frac{M_{\mathbf{K}}(\Pi)}{4}+1\geq n.$

######  Remark 10.

We note that unlike in Oler’s original inequality, equality in (3) of Theorem
3.6 is attained in a variety of ways. This is illustrated in Fig. 5, where the
polygon $\Pi$ consists of blocks of zig-zags and simple closed polygons having
sides parallel to the two sides of a chosen $\square(K)$.

Figure 5. A TS-packing of translates of $K$ (with $K$ being a circular disk
for the sake of simplicity), which satisfies the conditions in Theorem 3.6 and
for which there is equality in (3) of Theorem 3.6.

The following problem raised in [BeLa20] is still open.

######  Problem 4.

Characterize the case of equality in (3) of Theorem 3.6.

As an application of Theorem 3.6, we outline the proof of (3.4.2) of Theorem
3.4 for centrally symmetric $\mathbf{K}$ following [BeLa20]. It goes as
follows. Note that ${\rm bd}(\mathbf{C})$ satisfies the conditions in Theorem
3.6, and thus, we have

$\frac{{\rm area}(\mathbf{C})}{{\rm
area}\left(\square(\mathbf{K})\right)}+\frac{M_{\mathbf{K}}\left({\rm
bd}(\mathbf{C})\right)}{4}+1\geq n.$

Next, recall the following inequality from [BeLa20].

######  Lemma 3.7.

Let $\mathbf{K}$ be a convex domain in ${\mathbb{E}}^{2}$ and let $\mathbf{Q}$
be a convex polygon. Furthermore, let $A(\mathbf{Q},\mathbf{K})$ denote the
mixed area of $\mathbf{Q}$ and $\mathbf{K}$. If $\mathbf{K}$ is centrally
symmetric, then

$\frac{8A(\mathbf{Q},\mathbf{K})}{{\rm
area}\left(\square(\mathbf{K})\right)}\geq M_{\mathbf{K}}\left({\rm
bd}(\mathbf{Q})\right).$

Thus, Lemma 3.7 yields that

$\frac{{\rm area}(\mathbf{C})}{{\rm
area}\left(\square(\mathbf{K})\right)}+\frac{2A(\mathbf{C},\mathbf{K})}{{\rm
area}\left(\square(\mathbf{K})\right)}+1\geq n.$

From this, it follows that

${\rm area}\left({\rm
conv}\left(\bigcup_{i=1}^{n}(\mathbf{c}_{i}+\mathbf{K})\right)\right)={\rm
area}(\mathbf{C}+\mathbf{K})={\rm
area}(\mathbf{C})+2A(\mathbf{C},\mathbf{K})+{\rm area}(\mathbf{K})\geq$
$(n-1){\rm area}\left(\square(\mathbf{K})\right)+{\rm area}(\mathbf{K}),$

finishing the proof of (3.4.2) of Theorem 3.4 for centrally symmetric
$\mathbf{K}$.

###  3.2. Finding the densest totally separable packings of given number of
congruent spherical caps in the spherical plane

The Tammes problem, which is one of the best-known problems of discrete
geometry, was originally proposed by the Dutch botanist Tammes in 1930. It is
about finding the arrangement of $N$ points on the unit sphere
${\mathbb{S}}^{2}$ such that it maximizes the minimum distance between any two
points. Equivalently, for given $N>1$ one is asking for the largest $r$ such
that there exists a packing of $N$ spherical caps with angular radius $r$ in
the unit sphere ${\mathbb{S}}^{2}$. For a short overview of this problem see
[BL22] and for a comprehensive overview we refer the interested reader to the
recent article [Mu18].

The following definition is a natural extension to ${\mathbb{S}}^{2}$ of the
Euclidean analogue notion of Definition 3.

######  Definition 6.

A family of spherical caps of ${\mathbb{S}}^{2}$ is called a totally separable
packing in short, a TS-packing if any two spherical caps can be separated by a
great circle of ${\mathbb{S}}^{2}$ which is disjoint from the interior of each
spherical cap in the packing.

The analogue of the Tammes problem for TS-packings, called the separable
Tammes problem, was raised in [BL22].

######  Definition 7.

For given $k>1$ find the largest $r>0$ such that there exists a TS-packing of
$k$ spherical caps with angular radius $r$ in ${\mathbb{S}}^{2}$. Let us
denote this $r$ by $r_{\rm STam}(k,{\mathbb{S}}^{2})$.

######  Remark 11.

It was noted in [BL22] that $r_{\rm
STam}(2k^{\prime}-1,{\mathbb{S}}^{2})=r_{\rm
STam}(2k^{\prime},{\mathbb{S}}^{2})$ holds for any integer $k^{\prime}>1$.

The following TS-packings are special solutions of the separable Tammes
problem ([BL22]). Consider three mutually orthogonal great circles on
${\mathbb{S}}^{2}$. These divide the sphere into eight regular spherical
triangles of side lengths $\frac{\pi}{2}$. The family of inscribed spherical
caps of these triangles is a TS-packing of eight spherical caps of radius
$\arcsin\frac{1}{\sqrt{3}}(\approx 35.26^{\circ})$. We call such a family an
_octahedral TS-packing_ (Figure 7). Similarly, the side lines of a regular
spherical triangle of side length $\arccos\frac{1}{4}\approx 75.52^{\circ}$
divide the sphere into two regular spherical triangles of side length
$\arccos\frac{1}{4}$, and six isosceles spherical triangles of side lengths
$\pi-\arccos\frac{1}{4},\pi-\arccos\frac{1}{4}$ and $\arccos\frac{1}{4}$. The
inscribed spherical caps of the six isosceles triangles form a TS-packing of
six spherical caps of radius $\arctan\frac{3}{4}\approx 36.87^{\circ}$, which
we call a _cuboctahedral TS-packing_ (Figure 7).

Figure 6. An octahedral TS-packing in ${\mathbb{S}}^{2}$.

Figure 7. A cuboctahedral TS-packing in ${\mathbb{S}}^{2}$.

On the one hand, it is easy to check that $r_{\rm
STam}(2,{\mathbb{S}}^{2})=\frac{\pi}{2}(=90^{\circ})$ and $r_{\rm
STam}(3,{\mathbb{S}}^{2})=r_{\rm
STam}(4,{\mathbb{S}}^{2})=\frac{\pi}{4}(=45^{\circ})$. On the other hand,
Problem 7 is solved for $k=5,6,7,8$ in [BL22].

######  Definition 8.

Let $k>1$ be fixed. A TS-packing of $k$ spherical caps of radius $r_{\rm
STam}(k,{\mathbb{S}}^{2})$ is called _$k$ -optimal_.

######  Theorem 3.8.

For $5\leq k\leq 6$ we have $r_{\rm
STam}(k,{\mathbb{S}}^{2})=\arctan\frac{3}{4}$, and any $k$-optimal TS-packing
is a subfamily of a cuboctahedral TS-packing. Furthermore, for $7\leq k\leq 8$
we have $r_{\rm STam}(k,{\mathbb{S}}^{2})=\arcsin\frac{1}{\sqrt{3}}$, and any
$k$-optimal TS-packing is a subfamily of an octahedral TS-packing.

The value of $r_{\rm STam}(k,{\mathbb{S}}^{2})$ is not known for any $k>8$,
which leads to

######  Problem 5.

Compute the exact value of $r_{\rm STam}(k,{\mathbb{S}}^{2})$ for some small
integers $k>8$.

On the other hand, the authors [BL22] lower and upper bounded $r_{\rm
STam}(k,{\mathbb{S}}^{2})$ for any (sufficiently large value of) $k$ as
follows.

######  Theorem 3.9.

  * (i)

$r_{\rm
STam}(k,{\mathbb{S}}^{2})\leq\arccos\frac{1}{\sqrt{2}\sin\left(\frac{k}{k-2}\frac{\pi}{4}\right)}$
for all $k\geq 5$. In particular, $r_{\rm
STam}(8,{\mathbb{S}}^{2})=\arccos\sqrt{\frac{2}{3}}$
$=\arcsin\frac{1}{\sqrt{3}}(\approx 35.26^{\circ})$.

  * (ii)

For any sufficiently large value of $k$, we have $r_{\rm
STam}(k,{\mathbb{S}}^{2})\geq\frac{0.793}{\sqrt{k}}$.

Notice that the upper bound on $r_{\rm STam}(k,{\mathbb{S}}^{2})$ stated in
Part (i) is about $\sqrt{\frac{\pi}{k}}\approx\frac{1.772}{\sqrt{k}}$ as
$k\to+\infty$, which is of the same order of magnitude as the order of
magnitude of the lower bound given in Part (ii). Thus, one may wonder whether
the following question has a positive answer.

######  Problem 6.

Does the limit $\lim_{k\to+\infty}\sqrt{k}\cdot r_{\rm
STam}(k,{\mathbb{S}}^{2})$ exist?

###  3.3. On densest $\lambda$-separable circle packings in the plane of
constant curvature

A natural common generalization of packings and totally packings of circular
disks in planes of constant curvature was introduced in [AMC] as follows.

######  Definition 9.

Let $0\leq\lambda\leq\rho$. Let $\mathcal{P}$ be a packing of (circular) disks
of radius $\rho$ in
$\mathbb{M}\in\\{{\mathbb{E}}^{2},\mathbb{H}^{2},\mathbb{S}^{2}\\}$, where
$\mathbb{H}^{2}$ denotes the hyperbolic plane. We say that $\mathcal{P}$ is a
_$\lambda$ -separable packing of disks of radius $\rho$_ if the family
$\mathcal{P^{\prime}}$ of disks concentric to the disks of $\mathcal{P}$
having radius $\lambda$ form a totally separable packing in $\mathbb{M}$,
i.e., any two disks of $\mathcal{P^{\prime}}$ can be separated by a line in
$\mathbb{M}$ which is disjoint from the interior of every disk of
$\mathcal{P^{\prime}}$.

Note that a packing of disks of radius $\rho$ in $\mathbb{M}$ is a
$0$-separable packing of disks of radius $\rho$ in
$\mathbb{M}\in\\{{\mathbb{E}}^{2},\mathbb{H}^{2},\mathbb{S}^{2}\\}$. On the
other hand, a totally separable packing of disks of radius $\rho$ in
$\mathbb{M}$ is a $\rho$-separable packing of disks of radius $\rho$ in
$\mathbb{M}\in\\{{\mathbb{E}}^{2},\mathbb{H}^{2},\mathbb{S}^{2}\\}$.

It is a natural problem to investigate the maximum density of a
$\lambda$-separable packings of disks of radius $\rho$ in
$\mathbb{M}\in\\{{\mathbb{E}}^{2},\mathbb{H}^{2},\mathbb{S}^{2}\\}$. Before
stating the related results in [AMC], we introduce some notation that we need
in their formulation.

Let

(4)
$x_{1}^{s}(y):=\frac{1}{2}\arcsin\frac{\cos\lambda\sin^{2}y}{\sqrt{\sin^{2}y-\sin^{2}\lambda}},\hbox{
if }0<\lambda<\frac{\pi}{4},\arcsin\tan\lambda<y<\frac{\pi}{2},$ (5)
$x_{2}^{s}(y):=\frac{\pi}{2}-\frac{1}{2}\arcsin\frac{\cos\lambda\sin^{2}y}{\sqrt{\sin^{2}y-\sin^{2}\lambda}},\hbox{
if }0<\lambda<\frac{\pi}{4},\arcsin\tan\lambda<y<\frac{\pi}{2},$ (6)
$x^{h}(y):=\frac{1}{2}\operatorname{arcsinh}\frac{\cosh\lambda\sinh^{2}y}{\sqrt{\sinh^{2}y-\sinh^{2}\lambda}},\hbox{
if }0<\lambda<y,$ (7)
$x^{e}(y):=\frac{y^{2}}{2\sqrt{y^{2}-\lambda^{2}}},\hbox{ if }0<\lambda<y.$

Furthermore, for any
$0<\lambda<\frac{\pi}{4},\arcsin\tan\lambda<y<\frac{\pi}{2}$ and $i=1,2$ let
$T_{i}^{s}(y)$ denote the spherical isosceles triangle of edge lengths
$2y,2x_{i}^{s}(y),2x_{i}^{s}(y)$. Similarly, for any $0<\lambda<y$, let
$T^{h}(y)$ denote the hyperbolic isosceles triangle with edge lengths
$2y,2x^{h}(y),2x^{h}(y)$ and by $T^{e}(y)$ the Euclidean isosceles triangle
with edge lengths $2y,2x^{e}(y),2x^{e}(y)$. In addition, we denote the regular
spherical, hyperbolic, Euclidean triangle of edge length $2\rho$ by
$T_{reg}^{s}(\rho)$,$T_{reg}^{h}(\rho)$ and $T_{reg}^{e}(\rho)$, respectively.

######  Remark 12.

It is an elementary computation to check that $x_{1}^{s}$ as a function of $y$
is strictly decreasing over the closed interval
$S_{1}:=[\arcsin\tan\lambda,\arcsin(\sqrt{2}\sin\lambda)]$ and strictly
increasing over $S_{2}:=[\arcsin(\sqrt{2}\sin\lambda),\pi/2]$, with
$x_{1}^{s}(\arcsin\tan\lambda)=x_{1}^{s}(\pi/2)=\frac{\pi}{4}$ and
$x_{1}^{s}(\arcsin(\sqrt{2}\sin\lambda))=\lambda$. Since
$x_{2}^{s}(y)=\frac{\pi}{2}-x_{1}^{s}(y)$, similar statements hold for
$x_{2}^{s}$. For $k=1,2$ and $j=1,2$, we denote the inverse of the restriction
of $x_{j}^{s}$ to $S_{k}$ by $x_{j}^{s}|_{S_{k}}^{-1}$. We remark that
$\arcsin\tan\lambda<\arcsin(\sqrt{2}\sin\lambda)<\frac{\pi}{2}$ holds for all
$\lambda\in(0,\pi/4)$. Furthermore, $x^{h}(y)$ is a strictly decreasing
function of $y$ on
$H_{1}:=(\lambda,\operatorname{arcsinh}(\sqrt{2}\sinh\lambda)]$, and strictly
increasing on $H_{2}:=[\operatorname{arcsinh}(\sqrt{2}\sinh\lambda),+\infty)$,
with $x^{h}(y)\to+\infty$ as $y\to\lambda^{+}$ or $y\to+\infty$. We denote the
inverse of $x^{h}$ on the two intervals by $x^{h}|_{H_{1}}^{-1}$ and
$x^{h}|_{H_{2}}^{-1}$, respectively.

Furthermore, for any $0<\lambda\leq\arcsin\frac{3}{5}$, let

(8)
$y_{s}^{s}(\lambda):=\arcsin\sqrt{\frac{3+5\sin^{2}\lambda-\sqrt{9-34\sin^{2}\lambda+25\sin^{4}\lambda}}{8}}\hbox{
and}$ (9)
$y_{b}^{s}(\lambda):=\arcsin\sqrt{\frac{3+5\sin^{2}\lambda+\sqrt{9-34\sin^{2}\lambda+25\sin^{4}\lambda}}{8}},$

and for any $\lambda>0$, let

(10)
$y_{s}^{h}(\lambda):=\operatorname{arcsinh}\sqrt{\frac{5\sinh^{2}\lambda-3+\sqrt{25\sinh^{4}\lambda+34\sinh^{2}\lambda+9}}{8}}.$

An elementary computation shows that the above expressions exist on the
required intervals.

Finally, in the formulations of Theorems 3.10-3.12, by for any triangle $T$ in
$\mathbb{M}$ with
$\mathbb{M}\in\\{{\mathbb{E}}^{2},\mathbb{S}^{2},\mathbb{H}^{2}$}, $\delta(T)$
denotes the area ratio
$\frac{\operatorname{area}((\mathbf{B}_{1}\cup\mathbf{B}_{2}\cup\mathbf{B}_{3})\cap
T)}{\operatorname{area}(T)}$, where $\mathbf{B}_{1},\mathbf{B}_{2}$ and
$\mathbf{B}_{3}$ are disks of radius $\rho$ centered at the vertices of $T$.
In the case $\mathbb{M}={\mathbb{E}}^{2}$ we also assume that $\rho=1$.

######  Definition 10.

Let $\delta_{\lambda}$ (resp., $\delta_{\lambda}^{*}$) denote the largest
(upper) density of $\lambda$-separable packings of unit disks (resp.,
$\lambda$-separable lattice packings of unit disks) in ${\mathbb{E}}^{2}$.

It was proved in [FeFe] that $\delta_{1}=\delta_{1}^{*}=\frac{\pi}{4}$. On the
other hand, it is well known [Lagerungen] that
$\delta_{0}=\delta_{0}^{*}=\frac{\pi}{\sqrt{12}}$. The following theorem
extends these results to $\lambda$-separable packings as follows.

######  Theorem 3.10.

We have

$\delta_{\lambda}=\delta_{\lambda}^{*}=\left\\{\begin{array}[]{l}\frac{\pi}{\sqrt{12}},\hbox{
if }0\leq\lambda\leq\frac{\sqrt{3}}{2},\\\ \frac{\pi}{4\lambda},\hbox{ if
}\frac{\sqrt{3}}{2}\leq\lambda\leq 1.\\\ \end{array}\right.$

Furthermore, for $0\leq\lambda\leq\frac{\sqrt{3}}{2}$ and
$\frac{\sqrt{3}}{2}\leq\lambda\leq 1$ the lattice packing $\mathcal{F}$ of
unit disks whose Delaunay triangles are $T_{reg}^{e}(1)$ and
$T^{e}(\sqrt{2-2\sqrt{1-\lambda^{2}}})$, respectively, is a
$\lambda$-separable packing with density $\delta_{\lambda}$ (see Figure 8).

Figure 8. A densest $\lambda$-separable packing of unit disks in
${\mathbb{E}}^{2}$ with $\lambda=0.93$. The unit disks and the sides of the
Delaunay triangles are denoted by solid lines. The concentric disks of radius
$\lambda$ and the lines separating them are drawn with dotted and dashed
lines, respectively. The bases of the Delaunay triangles are horizontal, and
their length is greater than $2$. The legs of the Delaunay triangles are of
length $2$.

The next theorem deals with $\lambda$-separable packings of spherical caps of
radius $\rho$ in $\mathbb{S}^{2}$. If such a packing contains at least three
caps, then $\lambda\leq\frac{\pi}{4}$, and it can be shown that in this case
the inequality $\lambda\leq\frac{\pi}{2}-\rho$ is also satisfied.

######  Definition 11.

Let $\delta_{\lambda}^{s}(\rho)$ denote the largest density of
$\lambda$-separable packings of spherical caps of radius $\rho$ in
$\mathbb{S}^{2}$, i.e., the largest fraction of $\mathbb{S}^{2}$ covered by a
$\lambda$-separable packing of spherical caps of radius $\rho$ in
$\mathbb{S}^{2}$.

######  Theorem 3.11.

Let $0\leq\lambda\leq\rho<\frac{\pi}{2}$ with $\rho>0$ and
$\lambda\leq\frac{\pi}{2}-\rho$. Then

(11)
$\delta_{\lambda}^{s}(\rho)\leq\left\\{\begin{array}[]{l}\delta(T_{1}^{s}(x_{1}^{s}|_{S_{1}}^{-1}(\rho))),\hbox{
if
}\lambda\leq\rho\leq\min\left\\{y_{s}^{s}(\lambda),\frac{\pi}{4}\right\\},\\\
\delta(T_{reg}^{s}(\rho)),\hbox{ if }y_{s}^{s}(\lambda)<\rho\leq
y_{b}^{s}(\lambda),\\\ \delta(T_{2}^{s}(\rho)),\hbox{ if
}\frac{\pi}{4}<\rho,\hbox{ and }\rho<y_{s}^{s}(\lambda)\hbox{ or
}\rho>x_{b}^{s}(\lambda).\end{array}\right.$

Furthermore, in the above three cases we have equality if and only if there is
a $\lambda$-separable packing $\mathcal{F}$ of spherical caps of radius $\rho$
whose Delaunay triangles tile $\mathbb{S}^{2}$ and are congruent to
$T_{1}^{s}(x_{1}^{s}|_{S_{1}}^{-1}(\rho))$, $T_{reg}^{s}(\rho)$, or
$T_{2}^{s}(\rho)$, respectively.

Before the next definition we note that it has been observed by Böröczky
[Boroczky] that the notion of density of sphere packings in hyperbolic space
has to to be introduced with respect to well-defined underlying cell
decompositions of the hyperbolic space. The refined Molnár decomposition of a
packing, which is a variant the decomposition defined by Molnár in [Molnar]
for Euclidean circle packings, was introduced for the Euclidean, spherical and
hyperbolic plane in [BeLa20, BL22, AMC], respectively.

######  Definition 12.

Let $\delta_{\lambda}^{h}(\rho)$ denote the largest density of
$\lambda$-separable packings of hyperbolic disks of radius $\rho$ in the cells
of the hyperbolic refined Molnár decompositions in $\mathbb{H}^{2}$.

Our result for the density of $\lambda$-separable hyperbolic disks is as
follows.

######  Theorem 3.12.

Let $0\leq\lambda\leq\rho$, where $0<\rho$. Then

(12)
$\delta_{\lambda}^{h}(\rho)\leq\left\\{\begin{array}[]{l}\delta(T^{h}(x^{h}|_{H_{1}}^{-1}(\rho))),\hbox{
if }\lambda\leq\rho\leq y_{s}^{h}(\lambda),\\\
\delta(T_{reg}^{s}(\rho)),\hbox{ if }y_{s}^{h}(\lambda)<\rho.\\\
\end{array}\right.$

Furthermore, in the above two cases we have equality if there is a
$\lambda$-separable packing of hyperbolic disks of radius $\rho$ whose
Delaunay triangles are congruent to $T^{h}(x^{h}|_{H_{1}}^{-1}(\rho))$ or
$T_{reg}^{h}(\rho)$, respectively.

###  3.4. On the densest separable translative packings in Euclidean spaces of
dimensions greater than $\mathbf{2}$

It is natural to extend the concept of TS-packings discussed in Definition 3
to higher dimensions as follows.

######  Definition 13.

A packing $\mathcal{F}$ of convex bodies in ${\mathbb{E}}^{d}$, $d>2$ is
called a _totally separable packing_ , in short, a _TS-packing_ , if any two
members of $\mathcal{F}$ can be separated by a hyperplane of
${\mathbb{E}}^{d}$ which is disjoint from the interiors of all members of
$\mathcal{F}$.

An elegant paper of Kertész [Ke88] shows that the (upper) density of any TS-
packing of unit diameter balls in ${\mathbb{E}}^{3}$ is at most
$\frac{\pi}{6}$ with equality for the lattice packing of unit diameter balls
having integer coordinates. Actually, Kertész [Ke88] proved the following
stronger result: If a cube of volume $V>0$ in ${\mathbb{E}}^{3}$ contains a
TS-packing of $N>1$ balls or radius $r>0$, then $V\geq 8Nr^{3}$. In fact, it
is not hard to see that Kertész’s method of proof from [Ke88] implies the
following even stronger result.

######  Theorem 3.13.

If a cube of volume $V>0$ in ${\mathbb{E}}^{3}$ is partitioned into $N>1$
convex cells by N-1 successive plane cuts (just one cell being divided by each
cut) such that each convex cell contains a ball of radius $r>0$, then the sum
of the surface areas of the $N$ convex cells is at least $24Nr^{2}$ and
therefore $V\geq\frac{r}{3}\left(24Nr^{2}\right)=8Nr^{3}$.

It would be very interesting to find analogues of Theorem 3.13 in higher
dimensions.

######  Problem 7.

Prove or disprove that if a $d$-dimensional cube of volume $V>0$ in
${\mathbb{E}}^{d}$, $d>3$ is partitioned into $N>1$ convex cells by $N-1$
successive hyperplane cuts (just one cell being divided by each cut) such that
each convex cell contains a ball of radius $r>0$, then $V\geq 2^{d}Nr^{d}$.

######  Remark 13.

A positive answer to Problem 7 would imply that the (upper) density of any TS-
packing of unit diameter balls in ${\mathbb{E}}^{d}$, $d>3$ is at most
$\frac{\kappa_{d}}{2^{d}}$ with equality for the lattice packing of unit
diameter balls having integer coordinates in ${\mathbb{E}}^{d}$.

Next, recall the following elegant theorem of Böröczky Jr. [Bo94]: Consider
the convex hull $\mathbf{Q}$ of $n$ non-overlapping translates of an arbitrary
convex body $\mathbf{C}$ in $\mathbb{E}^{d}$ with $n$ being sufficiently
large. If $\mathbf{Q}$ has minimal mean $i$-dimensional projection for given
$i$ with $1\leq i<d$, then $\mathbf{Q}$ is approximately a $d$-dimensional
ball. The authors [BeLa19] proved an extension of this theorem to the so-
called $\rho$-separable translative packings of convex bodies in
$\mathbb{E}^{d}$. In short, one can regard $\rho$-separable packings (for
$\rho\geq 3$) as packings that are locally totally separable. In what follows,
we define the concept of $\rho$-separable translative packings using [BeLa19]
and then state the main result of [BeLa19].

######  Definition 14.

Let $\mathbf{C}$ be an $\mathbf{o}$-symmetric convex body of $\mathbb{E}^{d}$.
Furthermore, let $|\cdot|_{\mathbf{C}}$ denote the norm generated by
$\mathbf{C}$, i.e., let $|\mathbf{x}|_{\mathbf{C}}:=\inf\\{\lambda\ |\
\mathbf{x}\in\lambda\mathbf{C}\\}$ for any $\mathbf{x}\in\mathbb{E}^{d}$. Now,
let $\rho\geq 1$. We say that the packing

${\mathcal{P}}_{{\rm sep}}:=\\{\mathbf{c}_{i}+\mathbf{C}\ |\ i\in I\ {\rm
with}\ |\mathbf{c}_{j}-\mathbf{c}_{k}|_{\mathbf{C}}\geq 2\ {\rm for\ all}\
j\neq k\in I\\}$

of (finitely or infinitely many) non-overlapping translates of $\mathbf{C}$
with centers $\\{\mathbf{c}_{i}\ |\ i\in I\\}$ is a $\rho$-separable packing
in $\mathbb{E}^{d}$ if for each $i\in I$ the finite packing
$\\{\mathbf{c}_{j}+\mathbf{C}\ |\
\mathbf{c}_{j}+\mathbf{C}\subseteq\mathbf{c}_{i}+\rho\mathbf{C}\\}$ is a TS-
packing (in $\mathbf{c}_{i}+\rho\mathbf{C}$). Finally, let $\delta_{{\rm
sep}}(\rho,\mathbf{C})$ denote the largest upper density of all
$\rho$-separable translative packings of $\mathbf{C}$ in $\mathbb{E}^{d}$,
i.e., let

$\delta_{{\rm sep}}(\rho,\mathbf{C}):=\sup_{{\mathcal{P}}_{\rm
sep}}\left(\limsup_{\lambda\to+\infty}\frac{\sum_{\mathbf{c}_{i}+\mathbf{C}\subset\mathbf{W}_{\lambda}^{d}}{\rm
vol}_{d}(\mathbf{c}_{i}+\mathbf{C})}{{\rm
vol}_{d}(\mathbf{W}_{\lambda}^{d})}\right)\ ,$

where $\mathbf{W}_{\lambda}^{d}$ denotes the $d$-dimensional cube of edge
length $2\lambda$ centered at $\mathbf{o}$ in $\mathbb{E}^{d}$ having edges
parallel to the coordinate axes of $\mathbb{E}^{d}$.

######  Remark 14.

Let $\delta(\mathbf{C})$ (resp., $\delta_{{\rm sep}}(\mathbf{C})$) denote the
supremum of the upper densities of all translative packings (resp.,
translative TS-packings) of the $\mathbf{o}$-symmetric convex body
$\mathbf{C}$ in $\mathbb{E}^{d}$. Clearly, $\delta_{{\rm
sep}}(\mathbf{C})\leq\delta_{{\rm
sep}}(\rho,\mathbf{C})\leq\delta(\mathbf{C})$ for all $\rho\geq 1$.
Furthermore, if $1\leq\rho<3$, then any $\rho$-separable translative packing
of $\mathbf{C}$ in $\mathbb{E}^{d}$ is simply a translative packing of
$\mathbf{C}$ and therefore, $\delta_{{\rm
sep}}(\rho,\mathbf{C})=\delta(\mathbf{C})$.

Recall that the mean $i$-dimensional projection $M_{i}(\mathbf{C})$
($i=1,2,\ldots,d-1$) of the convex body $\mathbf{C}$ in $\mathbb{E}^{d}$, can
be expressed ([Sch14]) with the help of a mixed volume via the formula

$M_{i}(\mathbf{C})=\frac{\kappa_{i}}{\kappa_{d}}V(\overbrace{\mathbf{C},\ldots,\mathbf{C}}^{i},\overbrace{\mathbf{B}^{d},\ldots,\mathbf{B}^{d}}^{d-i}),$

where $\kappa_{d}$ is the volume of $\mathbf{B}^{d}$ in $\mathbb{E}^{d}$. Note
that $M_{i}(\mathbf{B}^{d})=\kappa_{i}$, and the surface volume of
$\mathbf{C}$ is
$\mathrm{svol}_{d-1}(\mathbf{C})=\frac{d\kappa_{d}}{\kappa_{d-1}}M_{d-1}(\mathbf{C})$
and in particular, $\mathrm{svol}_{d-1}(\mathbf{B}^{d})=d\kappa_{d}$. Set
$M_{d}(\mathbf{C}):={\rm vol}_{d}(\mathbf{C})$. Finally, recall that
$R(\mathbf{C})$ (resp., $r(\mathbf{C})$) denotes the circumradius (resp.,
inradius) of the convex body $\mathbf{C}$ in $\mathbb{E}^{d}$. The following
is the main result of [BeLa19].

######  Theorem 3.14.

Let $d\geq 2$, $1\leq i\leq d-1$, $\rho\geq 1$, and let $\mathbf{Q}$ be the
convex hull of a $\rho$-separable packing of $n$ translates of the
$\mathbf{o}$-symmetric convex body $\mathbf{C}$ in $\mathbb{E}^{d}$ such that
$M_{i}(\mathbf{Q})$ is minimal and $n\geq\frac{4^{d}d^{4d}}{\delta_{{\rm
sep}}(\rho,\mathbf{C})^{d-1}}\cdot\left(\rho\frac{R(\mathbf{C})}{r(\mathbf{C})}\right)^{d}$.
Then

(13) $\frac{r(\mathbf{Q})}{R(\mathbf{Q})}\geq
1-\frac{\omega}{n^{\frac{2}{d(d+3)}}},$

for $\omega=\lambda(d)\left(\frac{\rho
R(\mathbf{C})}{r(\mathbf{C})}\right)^{\frac{2}{d+3}}$, where $\lambda(d)$
depends only on the dimension $d$. In addition,

$M_{i}(\mathbf{Q})=\left(1+\frac{\sigma}{n^{\frac{1}{d}}}\right)M_{i}(\mathbf{B}^{d})\left(\frac{{\rm
vol}_{d}(\mathbf{C})}{\delta_{{\rm
sep}}(\rho,\mathbf{C})\kappa_{d}}\right)^{\frac{i}{d}}\cdot n^{\frac{i}{d}},$

where $-\frac{2.25R(\mathbf{C})\rho di}{r(\mathbf{C})\delta_{\rm
sep}(\rho,\mathbf{C})}\leq\sigma\leq\frac{2.1R(\mathbf{C})\rho
i}{r(\mathbf{C})\delta_{\rm sep}(\rho,\mathbf{C})}$.

######  Remark 15.

It is worth restating Theorem 3.14 as follows: Consider the convex hull
$\mathbf{Q}$ of $n$ non-overlapping translates of an arbitrary
$\mathbf{o}$-symmetric convex body $\mathbf{C}$ forming a $\rho$-separable
packing in $\mathbb{E}^{d}$ with $n$ being sufficiently large. If $\mathbf{Q}$
has minimal mean $i$-dimensional projection for given $i$ with $1\leq i<d$,
then $\mathbf{Q}$ is approximately a $d$-dimensional ball.

######  Problem 8.

Let $d\geq 2$, $1\leq i\leq d-1$, and let $\mathbf{C}$ be an
$\mathbf{o}$-symmetric convex body in $\mathbb{E}^{d}$. Does the analogue of
Theorem 3.14 hold for translative TS-packings of $\mathbf{C}$ in
$\mathbb{E}^{d}$?

######  Remark 16.

The nature of the question analogue to Theorem 3.14 on minimizing
$M_{d}(\mathbf{Q})={\rm vol}_{d}(\mathbf{Q})$ is very different. Namely,
recall that Betke and Henk [BeHe98] proved L. Fejes Tóth’s sausage conjecture
for $d\geq 42$ according to which the smallest volume of the convex hull of
$n$ non-overlapping unit balls in $\mathbb{E}^{d}$ is obtained when the $n$
unit balls form a sausage, that is, a linear packing. As linear packings of
unit balls are $\rho$-separable, therefore the above theorem of Betke and Henk
applies to $\rho$-separable packings of unit balls in $\mathbb{E}^{d}$ for all
$\rho\geq 1$ and $d\geq 42$.

We close this section with the following conjecture, which has already been
proved for $d=2$ (see (3.4.2) of Theorem 3.4) as well as for all $d\geq 42$
(see Remark 16).

######  Conjecture 3.

The volume of the convex hull of an arbitrary TS-packing of $N>1$ unit balls
in $\mathbb{E}^{d}$ with $3\leq d\leq 41$ is at least as large as the volume
of the convex hull of $N$ non-overlapping unit balls with their centers lying
on a line segment of length 2(N-1).

##  4\. Contact numbers for locally separable sphere packings

For an overview on contact graphs and contact numbers of packings we refer the
interested reader to the recent survey article [BeKh18]. Here we focus on the
latest developments for contact numbers of locally separable unit sphere
packings that are not discussed in [BeKh18].

Let us recall the definition of contact graphs and contact numbers for
packings.

######  Definition 15.

The contact graph $G(\mathcal{P})$ of a packing $\mathcal{P}$ of convex bodies
in ${\mathbb{E}}^{d}$, $d>1$ is the simple graph whose vertices correspond to
the members of the packing, and whose two vertices are connected by an edge if
the two members touch each other. The number of edges of $G(\mathcal{P})$ is
called the contact number $c(\mathcal{P})$ of $\mathcal{P}$.

The concept of locally separable (sphere) packing was introduced by the first
named author in [Bez21].

######  Definition 16.

We call the packing $\mathcal{P}$ of convex bodies in ${\mathbb{E}}^{d}$,
$d>1$ a locally separable packing, in short, an LS-packing if each member of
$\mathcal{P}$ together with the members of $\mathcal{P}$ that are tangent to
it form a TS-packing.

Clearly, any TS-packing is also an LS-packing, but not necessarily the other
way around. Moreover, it is worth noting that any $\rho$-separable packing by
translates of a convex body for $\rho=3$ is a translative LS-packing and vice
versa.

###  4.1. Largest contact numbers and crystallization for locally separable
unit disk packings

The following result was proved by the first named author in [Bez21], which is
an LS-packing analogue of the crystallization result of Heitmann and Radin
[HeRa] and characterizes all LS-packings of $n>1$ unit diameter disks having
maximum contact number.

######  Theorem 4.1.

Let $\mathcal{P}^{*}$ be an arbitrary LS-packing of $n>1$ unit diameter disks
in ${\mathbb{E}}^{2}$. Then

$c(\mathcal{P}^{*})\leq\lfloor 2n-2\sqrt{n}\rfloor,$

where $\lfloor\cdot\rfloor$ stands for the lower integer part of the
corresponding real number. Futhermore, suppose that $\mathcal{P}$ is an LS-
packing of $n$ unit diameter disks with $c(\mathcal{P})=\lfloor
2n-2\sqrt{n}\rfloor$, $n\geq 4$ in ${\mathbb{E}}^{2}$. Let
$G_{c}(\mathcal{P})$ denote the contact graph of $\mathcal{P}$ embedded in
${\mathbb{E}}^{2}$ such that the vertices are the center points of the unit
diameter disks of $\mathcal{P}$ and the edges are line segments of unit length
each connecting two center points if and only if the unit diameter disks
centered at those two points touch each other. Then either
$G_{c}(\mathcal{P})$ is the contact graph of the LS-packing of $7$ unit
diameter disks shown in Figure 9 or

(i) $G_{c}(\mathcal{P})$ is $2$-connected whose internal faces (i.e., faces
different from its external face) form an edge-to-edge connected family of
unit squares called a polyomino of an isometric copy of the integer lattice
${\mathbb{Z}}^{2}$ in ${\mathbb{E}}^{2}$ (see the first packing in Figure 9)
or

(ii) $G_{c}(\mathcal{P})$ is $2$-connected whose internal faces are unit
squares forming a polyomino of an isometric copy of the integer lattice
${\mathbb{Z}}^{2}$ in ${\mathbb{E}}^{2}$ with the exception of one internal
face which is a pentagon adjacent along (at least) three consecutive sides to
the external face of $G_{c}(\mathcal{P})$ and along (at most) two consecutive
sides to the polyomino (see the second packing in Figure 10) or

(iii) $G_{c}(\mathcal{P})$ possesses a degree one vertex on the boundary of
its external face such that deleting that vertex together with the edge
adjacent to it yields a $2$-connected graph whose internal faces are unit
squares forming a polyomino of an isometric copy of the integer lattice
${\mathbb{Z}}^{2}$ in ${\mathbb{E}}^{2}$ (see the first packing in Figure 10).

Figure 9. An LS-packing of $11$ (resp., $7$) unit diameter disks with maximum
contact number. Figure 10. Two LS-packings of $13$ unit diameter disks with
maximum contact number, the second of which is not a TS-packing.

The following crystallization result of TS-packings follows from Theorem 4.1
in a straightforward way.

######  Corollary 4.2.

Let $\mathcal{P}^{*}$ be an arbitrary TS-packing of $n>1$ unit diameter disks
in ${\mathbb{E}}^{2}$. Then

$c(\mathcal{P}^{*})\leq\lfloor 2n-2\sqrt{n}\rfloor.$

Futhermore, suppose that $\mathcal{P}$ is an TS-packing of $n$ unit diameter
disks with $c(\mathcal{P})=\lfloor 2n-2\sqrt{n}\rfloor$, $n\geq 4$ in
${\mathbb{E}}^{2}$. Then either $G_{c}(\mathcal{P})$ is the contact graph of
the LS-packing of $7$ unit diameter disks shown in Figure 9 or

(i) $G_{c}(\mathcal{P})$ is $2$-connected whose internal faces are unit
squares forming a polyomino of an isometric copy of the integer lattice
${\mathbb{Z}}^{2}$ in ${\mathbb{E}}^{2}$ (see the first packing in Figure 9)
or

(ii) $G_{c}(\mathcal{P})$ possesses a degree one vertex on the boundary of its
external face such that deleting that vertex together with the edge adjacent
to it yields a $2$-connected graph whose internal faces are unit squares
forming a polyomino of an isometric copy of the integer lattice
${\mathbb{Z}}^{2}$ in ${\mathbb{E}}^{2}$ (see the first packing in Figure 10).

###  4.2. Upper bounding the contact numbers of locally separable unit sphere
packings in dimensions larger than $\mathbf{2}$

In higher dimensions we know less than in the plane. At present the best upper
bound for contact numbers of LS-packings of congruent balls in
${\mathbb{E}}^{d}$, $d\geq 3$ is the one published by the first named author
in [Bez21]. We shall need the following notation. Let
$\mathcal{P}:=\\{\mathbf{B}^{d}[\mathbf{c}_{i},1]|i\in I\\}$ be an arbitrary
(finite or infinite) packing of unit balls in ${\mathbb{E}}^{d}$, $d\geq 3$
and let $\mathbf{V}_{i}:=\\{\mathbf{x}\in{\mathbb{E}}^{d}|\
|\mathbf{x}-\mathbf{c}_{i}|\leq|\mathbf{x}-\mathbf{c}_{j}|\ {\rm for\ all}\
j\neq i,j\in I\\}$ denote the Voronoi cell assigned to
$\mathbf{B}^{d}[\mathbf{c}_{i},1]$ for $i\in I$. Recall ([Rog]) that the
Voronoi cells $\\{\mathbf{V}_{i}|i\in I\\}$ form a face-to-face tiling of
${\mathbb{E}}^{d}$. Then let the largest density of the unit ball
$\mathbf{B}^{d}[\mathbf{c}_{i},1]$ in its truncated Voronoi cell
$\mathbf{V}_{i}\cap\mathbf{B}^{d}[\mathbf{c}_{i},\sqrt{d}]$ be denoted by
$\hat{\delta}_{d}$, i.e., let
$\hat{\delta}_{d}:=\sup_{\mathcal{P}}\left(\sup_{i\in I}\frac{\omega_{d}}{{\rm
vol}_{d}\left(\mathbf{V}_{i}\cap\mathbf{B}^{d}[\mathbf{c}_{i},\sqrt{d}]\right)}\right)$,
where $\mathcal{P}$ runs through all possible unit ball packings of
${\mathbb{E}}^{d}$. We are now ready to state the upper bound from [Bez21].

######  Theorem 4.3.

Let $\mathcal{P}$ be an arbitrary LS-packing of $n>1$ unit balls in
${\mathbb{E}}^{d}$, $d\geq 3$. Then

(14) $c(\mathcal{P})\leq\left\lfloor
dn-\left(d^{-\frac{d-3}{2}}\hat{\delta}_{d}^{-\frac{d-1}{d}}\right)n^{\frac{d-1}{d}}\right\rfloor.$

######  Remark 17.

Recall the following classical result of Rogers [Ro] (which was rediscovered
by Baranovskii [Ba] and extended to spherical and hyperbolic spaces by
Böröczky [Bo]): Let $\mathcal{P}:=\\{\mathbf{B}^{d}[\mathbf{c}_{i},1]|i\in
I\\}$ be an arbitrary packing of unit balls in ${\mathbb{E}}^{d}$, $d>1$ with
$\mathbf{V}_{i}$ standing for the Voronoi cell assigned to
$\mathbf{B}^{d}[\mathbf{c}_{i},1]$ for $i\in I$. Furthermore, take a regular
$d$-dimensional simplex of edge length $2$ in ${\mathbb{E}}^{d}$ and then draw
a $d$-dimensional unit ball around each vertex of the simplex. Finally, let
$\sigma_{d}$ denote the ratio of the volume of the portion of the simplex
covered by balls to the volume of the simplex. Then $\frac{\omega_{d}}{{\rm
vol}_{d}\left(\mathbf{V}_{i}\cap\mathbf{B}^{d}[\mathbf{c}_{i},\sqrt{\frac{2d}{d+1}}]\right)}\leq\sigma_{d}$
holds for all $i\in I$ and therefore $\hat{\delta}_{d}\leq\sigma_{d}$ for all
$d\geq 3$. The latter inequality and (14) yield that if $\mathcal{P}$ is an
arbitrary LS-packing of $n>1$ unit balls in ${\mathbb{E}}^{d}$, $d\geq 3$,
then $c(\mathcal{P})\leq\left\lfloor
dn-\left(d^{-\frac{d-3}{2}}\hat{\delta}_{d}^{-\frac{d-1}{d}}\right)n^{\frac{d-1}{d}}\right\rfloor\leq\left\lfloor
dn-\left(d^{-\frac{d-3}{2}}{\sigma}_{d}^{-\frac{d-1}{d}}\right)n^{\frac{d-1}{d}}\right\rfloor$,
where $\sigma_{d}\sim\frac{d}{e}2^{-\frac{1}{2}d}$ ([Ro]).

######  Remark 18.

We note that the density upper bound $\sigma_{d}$ of Rogers has been improved
by the first named author [Bez02] for dimensions $d\geq 8$ as follows: Using
the notations of Remark 17, [Bez02] shows that $\frac{\omega_{d}}{{\rm
vol}_{d}\left(\mathbf{V}_{i}\cap\mathbf{B}^{d}[\mathbf{c}_{i},\sqrt{\frac{2d}{d+1}}]\right)}\leq\hat{\sigma}_{d}$
holds for all $i\in I$ and $d\geq 8$ and therefore
$\hat{\delta}_{d}\leq\hat{\sigma}_{d}$ for all $d\geq 8$, where
$\hat{\sigma}_{d}$ is a geometrically well-defined quantity satisfying the
inequality $\hat{\sigma}_{d}<\sigma_{d}$ for all $d\geq 8$. This result and
(14) yield that if $\mathcal{P}$ is an arbitrary LS-packing of $n>1$ unit
balls in ${\mathbb{E}}^{d}$, $d\geq 8$, then $c(\mathcal{P})\leq\left\lfloor
dn-\left(d^{-\frac{d-3}{2}}\hat{\delta}_{d}^{-\frac{d-1}{d}}\right)n^{\frac{d-1}{d}}\right\rfloor\leq\left\lfloor
dn-\left(d^{-\frac{d-3}{2}}{\hat{\sigma}}_{d}^{-\frac{d-1}{d}}\right)n^{\frac{d-1}{d}}\right\rfloor$.

######  Remark 19.

The density upper bound $\sigma_{3}$ of Rogers has been improved by Hales [Ha]
as follows: If $\mathcal{P}:=\\{\mathbf{B}^{3}[\mathbf{c}_{i},1]|i\in I\\}$ is
an arbitrary packing of unit balls in ${\mathbb{E}}^{3}$ and $\mathbf{V}_{i}$
denotes the Voronoi cell assigned to $\mathbf{B}^{d}[\mathbf{c}_{i},1]$, $i\in
I$, then $\frac{\omega_{3}}{{\rm
vol}_{3}\left(\mathbf{V}_{i}\cap\mathbf{B}^{3}[\mathbf{c}_{i},\sqrt{2}]\right)}\leq\frac{\omega_{3}}{{\rm
vol}_{3}(\mathbf{D})}<0.7547<\sigma_{3}=0.7797...$, where $\mathbf{D}$ stands
for a regular dodacahedron of inradius $1$. Hence, $\hat{\delta}_{3}<0.7547$.
The latter inequality and (14) yield that if $\mathcal{P}$ is an arbitrary LS-
packing of $n>1$ unit balls in ${\mathbb{E}}^{3}$, then
$c(\mathcal{P})\leq\left\lfloor
3n-\hat{\delta}_{3}^{-\frac{2}{3}}n^{\frac{2}{3}}\right\rfloor\leq\left\lfloor
3n-1.206n^{\frac{2}{3}}\right\rfloor$.

######  Remark 20.

Let
$\mathcal{P}:=\\{\mathbf{B}^{d}[\mathbf{c}_{i},\frac{1}{2}]|\mathbf{c}_{i}\in{\mathbb{Z}}^{d},1\leq
i\leq n\\}$ be an arbitrary packing of $n$ unit diameter balls with centers
having integer coordinates in ${\mathbb{E}}^{d}$. Clearly, $\mathcal{P}$ is a
TS-packing. Then let $c_{{\mathbb{Z}}^{d}}(n)$ denote the largest
$c(\mathcal{P})$ for packings $\mathcal{P}$ of $n$ unit diameter balls of
${\mathbb{E}}^{d}$ obtained in this way. It is proved in [BeSzSz] that

(15) $dN^{d}-dN^{d-1}\leq c_{{\mathbb{Z}}^{d}}(n)\leq\left\lfloor dn-
dn^{\frac{d-1}{d}}\right\rfloor$

for $N\in{\mathbb{Z}}$ satisfying $0\leq N\leq n^{\frac{1}{d}}<N+1$, where
$d>1$ and $n>1$. Note that if $N=n^{\frac{1}{d}}\in{\mathbb{Z}}$, then the
lower and upper estimates of (15) are equal to $c_{{\mathbb{Z}}^{d}}(n)$.
Furthermore,

(16) $c_{{\mathbb{Z}}^{2}}(n)=\lfloor 2n-2\sqrt{n}\rfloor$

for all $n>1$. We note that [Ne] (resp., [AlCe]) generates an algorithm that
lists some (resp., all) packings
$\mathcal{P}=\\{\mathbf{B}^{d}[\mathbf{c}_{i},\frac{1}{2}]|\mathbf{c}_{i}\in{\mathbb{Z}}^{d},1\leq
i\leq n\\}$ with $c(\mathcal{P})=c_{{\mathbb{Z}}^{d}}(n)$ for $d\geq 4$
(resp., $d=2,3$) and $n>1$.

We cannot resist to close section with

######  Problem 9.

Let $\mathcal{P}$ be an arbitrary LS-packing (resp., TS-packing) of $n>1$ unit
diameter balls in ${\mathbb{E}}^{d}$, $d\geq 3$. Then prove or disprove that
$c(\mathcal{P})\leq c_{{\mathbb{Z}}^{d}}(n)$.

## Acknowledgements

Károly Bezdek was partially supported by a Natural Sciences and Engineering
Research Council of Canada Discovery Grant.

Zsolt Lángi was partially supported by the European Research Council Advanced
Grant “ERMiD”, and the NKFI research grant K147544.

## References


11institutetext:  Mir@cl Laboratory
University of Sfax, Tunisia
11email<EMAIL_ADDRESS>
# Abelian-square factors and binary words

Salah Triki

###### Abstract

In this work, we affirm the conjecture proposed by Gabriele Fici and Filippo
Mignosi in [1].

###### Keywords:

Abelian-square Factor Binary word

##  1 Definitions

###### Definition 1

A word $w$ is called a $factor$ of a word $u$ if there exists words x, y such
that $u=xwy$.

###### Definition 2

An abelian-square is a word of length $2n$ where the first $n$ symbols form an
anagram of the last $n$ symbols.

##  2 Conjecture

###### Lemma 1

Let $w$ is a word of length $n$, containing $k$ many distinct abelian-square
factors, and with the last symbol is in an abelian-square factor. Then a
binary word of length $n$ containing at least $k$ many distinct abelian-square
factors, and with the last symbol is in an abelian-square factor, exists.

The binary word will be called a binary image of $w$.

###### Proof

By induction on $n$. For $n<=2$, the claim is clear.
Assume that the claim holds for a word $w$ of length $n$ and $w^{\prime}$ is a
binary image of $w$. So, $wx$ with $x$ equals to the last symbol of $w$, has
$k$ many distinct abelian-square factors, and has a length $n+1$. And,
$w^{\prime}y$ with $y$ equals to the last symbol of $w^{\prime}$ is a binary
image of $wx$ ∎

###### Conjecture 1

[1] Assume that a word with length $n$, and containing $k$ many distinct
abelian-square factors, exists. Then a binary word of length $n$ containing at
least $k$ many distinct abelian-square factors exists.

###### Proof

By induction on $n$. For $n<=2$, the claim is clear.
Assume that the claim holds for a word $w$ of length $n$. So, if the last
symbol of $w$ is in a factor, then by lemma 1, $wx$ with $x$ equals to the
last symbol of $w$ has a binary image of length $n+1$ with at least $k$
distinct abelian-square factors. If the last symbol is not in a factor, then
also by lemma 1, $wx$ has a binary image with at least $k+1$ distinct abelian-
square factors

∎

## Acknowledgements

The author acknowledges, the financial support of this work from the Tunisian
General Direction of Scientific Research (DGRST).

## References

  * [1] Fici, G., Mignosi, F.: Words with the maximum number of abelian squares. CoRR abs/1506.03562 (2015), http://arxiv.org/abs/1506.03562