# On generalized Minkowski arrangements

Máté Kadlicskó  and  Zsolt Lángi∗ Department of Geometry, Budapest University
of Technology, Egry József utca 1., Budapest, Hungary, 1111
<EMAIL_ADDRESS>MTA-BME Morphodynamics Research Group and Department
of Geometry, Budapest University of Technology, Egry József utca 1., Budapest,
Hungary, 1111<EMAIL_ADDRESS>
###### Abstract.

The concept of a Minkowski arrangement was introduced by Fejes Tóth in 1965 as
a family of centrally symmetric convex bodies with the property that no member
of the family contains the center of any other member in its interior. This
notion was generalized by Fejes Tóth in 1967, who called a family of centrally
symmetric convex bodies a generalized Minkowski arrangement of order $\mu$ for
some $0<\mu<1$ if no member $K$ of the family overlaps the homothetic copy of
any other member $K^{\prime}$ with ratio $\mu$ and with the same center as
$K^{\prime}$. In this note we prove a sharp upper bound on the total area of
the elements of a generalized Minkowski arrangement of order $\mu$ of finitely
many circular disks in the Euclidean plane. This result is a common
generalization of a similar result of Fejes Tóth for Minkowski arrangements of
circular disks, and a result of Böröczky and Szabó about the maximum density
of a generalized Minkowski arrangement of circular disks in the plane. In
addition, we give a sharp upper bound on the density of a generalized
Minkowski arrangement of homothetic copies of a centrally symmetric convex
body.

###### Key words and phrases:

arrangement, Minkowski arrangement, density, homothetic copy

###### 2010 Mathematics Subject Classification:

52C15, 52C26, 52A10

∗ Corresponsing author

The second named author is supported by the National Research, Development and
Innovation Office, NKFI, K-119670, the János Bolyai Research Scholarship of
the Hungarian Academy of Sciences, and the BME IE-VIZ TKP2020 and ÚNKP-20-5
New National Excellence Programs by the Ministry of Innovation and Technology.

##  1\. Introduction

The notion of a _Minkowski arrangement_ of convex bodies was introduced by L.
Fejes Tóth in [7], who defined it as a family $\mathcal{F}$ of centrally
symmetric convex bodies in the $d$-dimensional Euclidean space
$\mathbb{R}^{d}$, with the property that no member of $\mathcal{F}$ contains
the center of any other member of $\mathcal{F}$ in its interior. He used this
concept to show, in particular, that the density of a Minkowski arrangement of
homothets of any given plane convex body with positive homogeneity is at most
four. Here an arrangement is meant to have positive homogeneity if the set of
the homothety ratios is bounded from both directions by positive constants. It
is worth mentioning that the above result is a generalization of the planar
case of the famous Minkowski Theorem from lattice geometry [12]. Furthermore,
Fejes Tóth proved in [7] that the density of a Minkowski arrangement of
circular disks in $\mathbb{R}^{2}$ with positive homogeneity is maximal for a
Minkowski arrangement of congruent circular disks whose centers are the points
of a hexagonal lattice and each disk contains the centers of six other members
on its boundary.

In [9], extending the investigation to finite Minkowski arrangements, Fejes
Tóth gave a sharp upper bound on the total area of the members of a Minkowski
arrangement of finitely many circular disks, and showed that this result
immediately implies the density estimate in [7] for infinite Minkowski circle-
arrangements. Following a different direction, in [8] for any $0<\mu<1$ Fejes
Tóth defined a _generalized Minkowski arrangements of order $\mu$_ as a family
$\mathcal{F}$ of centrally symmetric convex bodies with the property that for
any two distinct members $K,K^{\prime}$ of $\mathcal{F}$, $K$ does not overlap
the _$\mu$ -core_ of $K^{\prime}$, defined as the homothetic copy of
$K^{\prime}$ of ratio $\mu$ and concentric with $K^{\prime}$. In this paper he
made the conjecture that for any $0<\mu\leq\sqrt{3}-1$, the density of a
generalized Minkowski arrangement of circular disks with positive homogeneity
is maximal for a generalized Minkowski arrangement of congruent disks whose
centers are the points of a hexagonal lattice and each disk touches the
$\mu$-core of six other members of the family. According to [8], this
conjecture was verified by Böröczky and Szabó in a seminar talk in 1965,
though the first written proof seems to be published only in [5] in 2002. It
was observed both in [8] and [5] that if $\sqrt{3}-1<\mu<1$, then, since the
above hexagonal arrangement does not cover the plane, that arrangement has no
maximal density.

In this paper we prove a sharp estimate on the total area of a generalized
Minkowski arrangement of finitely many circular disks, with a characterization
of the equality case. Our result includes the result in [9] as a special case,
and immediately implies the one in [5]. The proof of our statement relies on
tools from both [5, 9], but uses also some new ideas. In addition, we also
generalize a result from Fejes Tóth [7] to find a sharp upper bound on the
density of a generalized Minkowski arrangement of homothetic copies of a
centrally symmetric convex body.

For completeness, we mention that similar statements for (generalized)
Minkowski arrangements in other geometries and in higher dimensional spaces
were examined, e.g. in [6, 10, 14]. Minkowski arrangements consisting of
congruent convex bodies were considered in [4]. Estimates for the maximum
cardinality of mutually intersecting members in a (generalized) Minkowski
arrangement can be found in [11, 15, 16, 18]. The problem investigated in this
paper is similar in nature to those dealing with the volume of the convex hull
of a family of convex bodies, which has a rich literature. This includes a
result of Oler [17] (see also [3]), which is also of lattice geometric origin
[21], and the notion of parametric density of Betke, Henk and Wills [1]. In
particular, our problem is closely related to the notion of density with
respect to outer parallel domains defined in [3]. Applications of
(generalized) Minkowski arrangements in other branches of mathematics can be
found in [19, 20].

As a preliminary observation, we start with the following generalization of
Remark 2 of [7], stating the same property for (not generalized) Minkowski
arrangements of plane convex bodies. In Proposition 1, by
$\operatorname{vol}_{d}(\cdot)$ we denote $d$-dimensional volume, and by
$\mathbf{B}^{d}$ we denote the closed Euclidean unit ball centered at the
origin.

######  Proposition 1.

Let $0<\mu<1$, let $K\subset\mathbb{R}^{d}$ be an origin-symmetric convex body
and let $\mathcal{F}=\\{x_{1}+\lambda_{1}K,x_{2}+\lambda_{2}K,\dots\\}$ be a
generalized Minkowski arrangement of order $\mu$, where
$x_{i}\in\mathbb{R}^{d},\lambda_{i}>0$ for each $i=1,2,\dots$. Assume that
$\mathcal{F}$ is of positive homogeneity, that is, there are constants
$0<C_{1}<C_{2}$ satisfying $C_{1}\leq\lambda_{i}\leq C_{2}$ for all values of
$i$, and define the (upper) density $\delta(\mathcal{F})$ of $\mathcal{F}$ in
the usual way as

$\delta(\mathcal{F})=\limsup_{R\to\infty}\frac{\sum_{x_{i}\in
R\mathbf{B}^{d}}\operatorname{vol}_{d}(x_{i}+\lambda_{i}K)}{\operatorname{vol}_{d}(R\mathbf{B}^{d})},$

if it exists. Then

(1) $\delta(\mathcal{F})\leq\frac{2^{d}}{(1+\mu)^{d}},$

where equality is attained, e.g. if $\\{x_{1},x_{2},\ldots\\}$ is a lattice
with $K$ as its fundamental region, and $\lambda_{i}=2/(1+\mu)$ for all values
of $i$.

###### Proof.

Note that the equality part of Proposition 1 clearly holds, and thus, we prove
only the inequality in (1). Let
$||\cdot||_{K}:\mathbb{R}^{d}\rightarrow[0,\infty)$ denote the norm with $K$
as its unit ball. Then, by the definition of a generalized Minkowski
arrangement, we have

$||x_{i}-x_{j}||_{K}\geq\max\\{\lambda_{i}+\mu\lambda_{j},\lambda_{j}+\mu\lambda_{i}\\}\geq$
$\geq\frac{1}{2}\left((\lambda_{i}+\mu\lambda_{j})+(\lambda_{j}+\mu\lambda_{i})\right)=\frac{1+\mu}{2}(\lambda_{i}+\lambda_{j}),$

implying that the homothets $x_{i}+(\lambda_{i}/2)\cdot\left(1+\mu\right)K$
are pairwise non-overlapping. In other words, the family
$\mathcal{F}^{\prime}=\left\\{x_{i}+(\lambda_{i}/2)\cdot\left(1+\mu\right)K:i=1,2,\ldots\right\\}$
is a packing. Thus, the density of $\mathcal{F}^{\prime}$ is at most one, from
which (1) readily follows. Furthermore, if $K$ is the fundamental region of a
lattice formed by the $x_{i}$’s and $\lambda_{i}=2/(1+\mu)$ for all values of
$i$, then $\mathcal{F}^{\prime}$ is a tiling, implying the equality case. ∎

Following the terminology of Fejes Tóth in [9] and to permit a simpler
formulation of our main result, in the remaining part of the paper we consider
generalized Minkowski arrangements of _open_ circular disks, where we note
that generalized Minkowski arrangements can be defined for families of open
circular disks in the same way as for families of closed circular disks.

To state our main result, we need some preparation, where we denote the
boundary of a set by $\operatorname{bd}(\cdot)$. Consider some generalized
Minkowski arrangement
$\mathcal{F}=\\{B_{i}=x_{i}+\rho_{i}\operatorname{int}(\mathbf{B}^{2}):i=1,2,\ldots,n\\}$
of open circular disks in $\mathbb{R}^{2}$ of order $\mu$, where $0<\mu<1$.
Set $U(\mathcal{F})=\bigcup_{i=1}^{n}B_{i}=\bigcup\mathcal{F}$. Then each
circular arc $\Gamma$ in $\operatorname{bd}(U(\mathcal{F}))$ corresponds to a
circular sector, which can be obtained as the union of the segments connecting
a point of $\Gamma$ to the center of the disk in $\mathcal{F}$ whose boundary
contains $\Gamma$. We call the union of these circular sectors the _outer
shell_ of $\mathcal{F}$. Now consider a point
$p\in\operatorname{bd}(U(\mathcal{F}))$ belonging to at least two members of
$\mathcal{F}$, say $B_{i}$ and $B_{j}$, such that $x_{i},x_{j}$ and $p$ are
not collinear. Assume that the convex angular region bounded by the two closed
half lines starting at $p$ and passing through $x_{i}$ and $x_{j}$,
respectively, do not contain the center of another element of $\mathcal{F}$ in
its interior which contains $p$ on its boundary. We call the union of the
triangles $\operatorname{conv}\\{p,x_{i},x_{j}\\}$ satisfying these conditions
the _inner shell_ of $\mathcal{F}$. We denote the inner and the outer shell of
$\mathcal{F}$ by $I(\mathcal{F})$ and $O(\mathcal{F})$, respectively. Finally,
we call the set $C(\mathcal{F})=U(\mathcal{F})\setminus(I(\mathcal{F})\cup
O(\mathcal{F}))$ the _core_ of $\mathcal{F}$ (cf. Figure 1). Clearly, the
outer shell of any generalized Minkowski arrangement of open circular disks is
nonempty, but there are arrangements for which $I(\mathcal{F})=\emptyset$ or
$C(\mathcal{F})=\emptyset$.

Figure 1. The outer and inner shell, and the core of an arrangement, shown in
white, light grey and dark grey, respectively.

If the intersection of two members of $\mathcal{F}$ is nonempty, then we call
this intersection a _digon_. If a digon touches the $\mu$-cores of both disks
defining it, we call the digon _thick_. A digon which is not contained in a
third member of $\mathcal{F}$ is called a _free digon_. Our main theorem is as
follows, where $\operatorname{area}(X)$ denotes the area of the set $X$.

######  Theorem 1.

Let $0<\mu\leq\sqrt{3}-1$, and let
$\mathcal{F}=\\{B_{i}=x_{i}+\rho_{i}\operatorname{int}(\mathbf{B}^{2}):i=1,2,\ldots,n\\}$
be a generalized Minkowski arrangement of finitely many open circular disks of
order $\mu$. Then

$T=\pi\sum_{i=1}^{n}\rho_{i}^{2}\leq\frac{2\pi}{\sqrt{3}(1+\mu)^{2}}\operatorname{area}(C(\mathcal{F}))+$
$+\frac{4\cdot\arccos({\frac{1+\mu}{2})}}{(1+\mu)\cdot\sqrt{(3+\mu)(1-\mu)}}\operatorname{area}(I(\mathcal{F}))+\operatorname{area}(O(\mathcal{F})),$

where $T$ is the total area of the circles, with equality if and only if each
free digon in $\mathcal{F}$ is thick.

In the paper, for any points $x,y,z\in\mathbb{R}^{2}$, we denote by $[x,y]$
the closed segment with endpoints $x,y$, by $[x,y,z]$ the triangle
$\operatorname{conv}\\{x,y,z\\}$, by $|x|$ the Euclidean norm of $x$, and if
$x$ and $z$ are distinct from $y$, by $\angle xyz$ we denote the measure of
the angle between the closed half lines starting at $y$ and passing through
$x$ and $z$. Note that according to our definition, $\angle xyz$ is at most
$\pi$ for any $x,z\neq y$. For brevity we call an open circular disk a _disk_
, and a generalized Minkowski arrangement of disks of order $\mu$ a _$\mu$
-arrangement_. Throughout Sections 2 and 3 we assume that
$0<\mu\leq\sqrt{3}-1$.

In Section 2, we prove some preliminary lemmas. In Section 3, we prove Theorem
1. Finally, in Section 4, we collect additional remarks and questions.

##  2\. Preliminaries

For any $B_{i},B_{j}\in\mathcal{F}$, if ${B_{i}\cap B_{j}}\neq\emptyset$, we
call the two intersection points of $\operatorname{bd}(B_{i})$ and
$\operatorname{bd}(B_{j})$ the _vertices_ of the digon $B_{i}\cap B_{j}$.

First, we recall the following lemma of Fejes Tóth [9, Lemma 2]. To prove it,
we observe that for any $\mu>0$, a generalized Minkowski arrangement of order
$\mu$ is a Minkowski arrangement as well.

######  Lemma 1.

Let $B_{i},B_{j},B_{k}\in\mathcal{F}$ such that the digon $B_{i}\cap B_{j}$ is
contained in $B_{k}$. Then the digon $B_{i}\cap B_{k}$ is free (with respect
to $\mathcal{F}$).

From now on, we call the maximal subfamilies $\mathcal{F}^{\prime}$ of
$\mathcal{F}$ (with respect to containment) with the property that
$\bigcup_{B_{i}\in\mathcal{F}^{\prime}}B_{i}$ is connected the _connected
components_ of $\mathcal{F}$. Our next lemma has been proved by Fejes Tóth in
[9] for Minkowski arrangements of order $\mu=0$. His argument can be applied
to prove Lemma 2 for an arbitrary value of $\mu$. Here we include this proof
for completeness.

######  Lemma 2.

If $\mathcal{F}^{\prime}$ is a connected component of $\mathcal{F}$ in which
each free digon is thick, then the elements of $\mathcal{F}^{\prime}$ are
congruent.

###### Proof.

We need to show that for any $B_{i},B_{j}\in\mathcal{F}^{\prime}$, $B_{i}$ and
$B_{j}$ are congruent. Observe that by connectedness, we may assume that
$B_{i}\cap B_{j}$ is a digon. If $B_{i}\cap B_{j}$ is free, then it is thick,
which implies that $B_{i}$ and $B_{j}$ are congruent. If $B_{i}\cap B_{j}$ is
not free, then there is a disk $B_{k}\in\mathcal{F}^{\prime}$ containing it.
By Lemma 1, the digons $B_{i}\cap B_{k}$ and $B_{j}\cap B_{k}$ are free. Thus
$B_{k}$ is congruent to both $B_{i}$ and $B_{j}$. ∎

In the remaining part of Section 2, we examine densities of some circular
sectors in certain triangles. The computations in the proofs of these lemmas
were carried out by a Maple 18.00 software.

######  Lemma 3.

Let $0<\gamma<\pi$ and $A,B>0$ be arbitrary. Let $T=[x,y,z]$ be a triangle
such that $\angle xzy=\gamma$, and $|x-z|=A$ and $|y-z|=B$. Let
$\Delta=\Delta(\gamma,A,B)$, $\alpha=\alpha(\gamma,A,B)$ and
$\beta=\beta(\gamma,A,B)$ denote the functions with variables $\gamma,A,B$
whose values are the area and the angles of $T$ at $x$ and $y$, respectively,
and set $f_{A,B}(\gamma)=\left(\alpha A^{2}+\beta B^{2}\right)/{\Delta}$.
Then, for any $A,B>0$, the function $f_{A,B}(\gamma)$ is strictly decreasing
on the interval $\gamma\in(0,\pi)$.

$B$$A$$\gamma$$z$$\beta$$y$$\alpha$$x$ Figure 2. Notation in Lemma 3.

###### Proof.

Without loss of generality, assume that $A\leq B$, and let $g=\alpha
A^{2}+\beta B^{2}$. Then, by an elementary computation, we have that

$g=A^{2}\operatorname{arccot}\frac{A-B\cos\gamma}{B\sin\gamma}+B^{2}\operatorname{arccot}\frac{B-A\cos\gamma}{A\sin\gamma},\hbox{
and }\Delta=\frac{1}{2}AB\sin\gamma.$

We regard $g$ and $\Delta$ as functions of $\gamma$. We intend to show that
$g^{\prime}\Delta-g\Delta^{\prime}$ is negative on the interval $(0,\pi)$ for
all $A,B>0$. Let $h=g^{\prime}\cdot\Delta/{\Delta^{\prime}}-g$, and note that
this expression is continuous on $\left(0,{\pi}/{2}\right)$ and
$\left({\pi}/{2},\pi\right)$ for all $A,B>0$. By differentiating and
simplifying, we obtain

$h^{\prime}=\frac{-2\left(A^{2}(1+\cos^{2}(\gamma))+B^{2}(1+\cos^{2}(\gamma))-4AB\cos(\gamma)\right)A^{2}B^{2}\sin^{2}(\gamma)}{\cos^{2}\left(\gamma)(A^{2}+B^{2}-2AB\cos(\gamma)\right)^{2}},$

which is negative on its domain. This implies that
$g^{\prime}\Delta-g\Delta^{\prime}$ is strictly decreasing on
$\left(0,{\pi}/{2}\right)$ and strictly increasing on
$\left({\pi}/{2},\pi\right)$. On the other hand, we have $\lim_{\gamma\to
0^{+}}\left(g^{\prime}\Delta-g\Delta^{\prime}\right)=-A^{3}B\pi$, and
$\lim_{\gamma\to\pi^{-}}\left(g^{\prime}\Delta-g\Delta^{\prime}\right)=0$.
This yields the assertion. ∎

######  Lemma 4.

Consider two disks $B_{i},B_{j}\in\mathcal{F}$ such that
$|x_{i}-x_{j}|<\rho_{i}+\rho_{j}$, and let $v$ be a vertex of the digon
$B_{i}\cap B_{j}$. Let $T=[x_{i},x_{j},v]$, $\Delta=\operatorname{area}(T)$,
and let $\alpha_{i}=\angle vx_{i}x_{j}$ and $\alpha_{j}=\angle vx_{j}x_{i}$.
Then

(2)
$\frac{1}{2}\alpha_{i}\rho_{i}^{2}+\frac{1}{2}\alpha_{j}\rho_{j}^{2}\leq\frac{4\arccos\frac{1+\mu}{2}}{(1+\mu)\sqrt{(1-\mu)(3+\mu)}}\Delta,$

with equality if and only if $\rho_{i}=\rho_{j}$ and
$|x_{i}-x_{j}|=\rho_{i}(1+\mu)$.

$v$$\alpha_{i}$$x_{i}$$\alpha_{j}$$x_{j}$$\rho_{i}$$\rho_{j}$ Figure 3.
Notation in Lemma 4.

###### Proof.

First, an elementary computation shows that if $\rho_{i}=\rho_{j}$ and
$|x_{i}-x_{j}|=\rho_{i}(1+\mu)$, then there is equality in (2).

Without loss of generality, let $\rho_{i}=1$, and $0<\rho_{j}=\rho\leq 1$. By
Lemma 3, we may assume that $|x_{i}-x_{j}|=1+\mu\rho$. Thus, the side lengths
of $T$ are $1,\rho,1+\mu\rho$. Applying the Law of Cosines and Heron’s formula
to $T$ we obtain that

$\frac{\frac{1}{2}\alpha_{i}\rho_{i}^{2}+\frac{1}{2}\alpha_{j}\rho_{j}^{2}}{\Delta}=\frac{f(\rho,\mu)}{g(\rho,\mu)},$

where

$f(\rho,\mu)=\frac{1}{2}\arccos\frac{1+(1+\mu\rho)^{2}-r^{2}}{2(1+\mu\rho)}+\frac{1}{2}\rho^{2}\arccos\frac{\rho^{2}+(1+\mu\rho)^{2}-1}{2\rho(1+\mu\rho)},$

and

$g(\rho,\mu)=\rho\sqrt{2+\rho+\mu\rho)(2-\rho+\mu\rho)(1-\mu^{2})}.$

In the remaining part we show that

$\frac{f(\rho,\mu)}{g(\rho,\mu)}<\frac{4\arccos\frac{1+\mu}{2}}{(1+\mu)\sqrt{(1-\mu)(3+\mu)}}$

if $0<\rho<1$ and $0\leq\mu\leq\sqrt{3}-1$. To do it we distinguish two
separate cases.

_Case 1_ , $0<\rho\leq 1/5$. In this case we estimate
$f(\rho,\mu)/g(\rho,\mu)$ as follows. Let the part of $[x_{i},x_{j}]$ covered
by both disks $B_{i}$ and $B_{j}$ be denoted by $S$. Then $S$ is a segment of
length $(1-\mu)\rho$. On the other hand, if $A_{i}$ denotes the convex
circular sector of $B_{i}$ bounded by the radii $[x_{i},v]$ and
$[x_{i},x_{j}]\cap B_{i}$, and we define $A_{j}$ analogously, then the sets
$A_{i}\cap A_{j}$ and $(A_{i}\cup A_{j})\setminus T$ are covered by the
rectangle with $S$ as a side which contains $v$ on the side parallel to $S$.
The area of this rectangle is twice the area of the triangle
$\operatorname{conv}(S\cup\\{v\\})$, implying that

$\frac{f(\rho,\mu)}{g(\rho,\mu)}\leq 1+\frac{2(1-\mu)\rho}{1+\mu\rho}.$

We show that if $0<\rho\leq 1/5$, then the right-hand side quantity in this
inequality is strictly less than the right-hand side quantity in (2). By
differentiating with respect to $\rho$, we see that as a function of $\rho$,
$1+\left(2(1-\mu)\rho\right)/(1+\mu\rho)$ is strictly increasing on its domain
and attains its maximum at $\rho=1/5$. Thus, using the fact that this maximum
is equal to $(7-\mu)/(5+\mu)$, we need to show that

$\frac{4\arccos\frac{1+\mu}{2}}{(1+\mu)\sqrt{(1-\mu)(3+\mu)}}-\frac{7-\mu}{5+\mu}>0.$

Clearly, the function

$\mu\mapsto\frac{\arccos\frac{1+\mu}{2}}{\frac{1+\mu}{2}}$

is strictly decreasing on the interval $[0,\sqrt{3}-1]$. By differentiation
one can easily check that the function

$\mu\mapsto\frac{7-\mu}{5+\mu}\sqrt{(1-\mu)(3+\mu)}$

is also strictly increasing on the same interval. Thus, we obtain that the
above expression is minimal if $\mu=\sqrt{3}-1$, implying that it is at least
$0.11570\ldots$.

_Case 2_ , $1/5<\rho\leq 1$.

We show that in this case the partial derivative
$\partial_{\rho}\left(f(\rho,\mu)/g(\rho,\mu)\right)$, or equivalently, the
quantity
$h(\rho,\mu)=f^{\prime}_{\rho}(\rho,\mu)g(\rho,\mu)-g^{\prime}_{\rho}(\rho,\mu)f(\rho,\mu)$
is strictly positive. By plotting the latter quantity on the rectangle
$0\leq\mu\leq\sqrt{3}-1$, $1/5\leq\rho\leq 1$, its minimum seems to be
approximately $0.00146046085$. To use this fact, we upper bound the two
partial derivatives of this function, and compute its values on a grid. In
particular, using the monotonicity properties of the functions $f,g$, we
obtain that under our conditions $|f(\rho,\mu)|<1.25$ and $|g(\rho,\mu)|\leq
0.5$. Furthermore, using the inequalities $0\leq\mu\leq\sqrt{3}-1$,
$1/5\leq\rho\leq 1$ and also the triangle inequality to estimate the
derivatives of $f$ and $g$, we obtain that

$|f^{\prime}_{\rho}(\rho,\mu)|<1.95,\,|f^{\prime}_{\mu}(\rho,\mu)|<2.8,\,|f^{\prime\prime}_{\rho\rho}(\rho,\mu)|<2.95,\,|f^{\prime\prime}_{\rho\mu}(\rho,\mu)|<9.8,$

and

$|g^{\prime}_{\rho}(\rho,\mu)|<0.93,\,|g^{\prime}_{\mu}(\rho,\mu)|<1.08,\,|g^{\prime\prime}_{\rho\rho}(\rho,\mu)|<2.64,\,|g^{\prime\prime}_{\rho\mu}(\rho,\mu)|<15.1.$

These inequalities imply that $|h^{\prime}_{\rho}(\rho,\mu)|<4.78$ and
$|h^{\prime}_{\mu}(\rho,\mu)|<28.49$, and hence, for any $\Delta_{\rho}$ and
$\Delta_{\mu}$, we have
$h(\rho+\Delta_{\rho},\mu+\Delta_{\mu})>h(\rho,\mu)-4.78|\Delta_{\rho}|-28.49|\Delta_{\mu}|$.
Thus, we divided the rectangle $[0.2,1]\times[0,\sqrt{3}-1]$ into a
$8691\times 8691$ grid, and by numerically computing the value of
$h(\rho,\mu)$ at the gridpoints, we showed that at any such point the value of
$h$ (up to $12$ digits) is at least $0.00144$. According to our estimates
above, this implies that $h(\rho,\mu)\geq 0.00002$ for all values of $\rho$
and $\mu$. ∎

Before our next lemma, recall that $\mathbf{B}^{2}$ denotes the _closed_ unit
disk centered at the origin.

######  Lemma 5.

For some $0<\nu<1$, let $x,y,z\in\mathbb{R}^{2}$ be non-collinear points, and
let $\\{B_{u}=u+\rho_{u}\mathbf{B}^{2}:u\in\\{x,y,z\\}\\}$ be a
$\nu$-arrangement of disks; that is, assume that for any
$\\{u,v\\}\subset\\{x,y,z\\}$, we have
$|u-v|\geq\max\\{\rho_{u},\rho_{v}\\}+\nu\min\\{\rho_{u},\rho_{v}\\}$. Assume
that for any $\\{u,v\\}\subset\\{x,y,z\\}$, $B_{u}\cap B_{v}\neq\emptyset$,
and that the union of the three disks covers the triangle $[x,y,z]$. Then
$\nu\leq\sqrt{3}-1$.

$x$$y$$z$$\rho_{j}$$\rho_{x}$$\rho_{y}$$\rho_{z}$$T$ Figure 4. Notation in
Lemma 5. The circles drawn with dotted lines represent the $\mu$-cores of the
disks.

###### Proof.

Without loss of generality, assume that $0<\rho_{z}\leq\rho_{y}\leq\rho_{x}$.
Since the disks are compact sets, by the Knaster-Kuratowski-Mazurkiewicz lemma
[13], there is a point $q$ of $T$ belonging to all the disks, or in other
words, there is some point $q\in T$ such that $|q-u|\leq\rho_{u}$ for any
$u\in\\{x,y,z\\}$. Recalling the notation $T=[x,y,z]$ from the introduction,
let $T^{\prime}=[x^{\prime},y^{\prime},z^{\prime}]$ be a triangle with edge
lengths $|y^{\prime}-x^{\prime}|=\rho_{x}+{\nu}\rho_{y}$,
$|z^{\prime}-x^{\prime}|=\rho_{x}+{\nu}\rho_{z}$ and
$|z^{\prime}-y^{\prime}|=\rho_{y}+{\nu}\rho_{z}$, and note that these lengths
satisfy the triangle inequality.

We show that the disks $x^{\prime}+\rho_{x}\mathbf{B}^{2}$,
$y^{\prime}+\rho_{y}\mathbf{B}^{2}$ and $z^{\prime}+\rho_{z}\mathbf{B}^{2}$
and $T^{\prime}$ satisfy the conditions in the lemma. To do this, we show the
following, more general statement, which, together with the trivial
observation that any edge of $T^{\prime}$ is covered by the two disks centered
at its endpoints, clearly implies what we want: For any triangles $T=[x,y,z]$
and $T^{\prime}=[x^{\prime},y^{\prime},z^{\prime}]$ satisfying
$|u^{\prime}-v^{\prime}|\leq|u-v|$ for any $u,v\in\\{x,y,z\\}$, and for any
point $q\in T$ there is a point $q^{\prime}\in T^{\prime}$ such that
$|q^{\prime}-u^{\prime}|\leq|q-u|$ for any $u\in\\{x,y,z\\}$. The main tool in
the proof of this statement is the following straightforward consequence of
the Law of Cosines, stating that if the side lengths of a triangle are
$A,B,C$, and the angle of the triangle opposite of the side of length $C$ is
$\gamma$, then for any fixed values of $A$ and $B$, $C$ is a strictly
increasing function of $\gamma$ on the interval $(0,\pi)$.

To apply it, observe that if we fix $x,y$ and $q$, and rotate $[x,z]$ around
$x$ towards $[x,q]$, we strictly decrease $|z-y|$ and $|z-q|$ and do not
change $|y-x|,|z-x|,|x-q|$ and $|y-q|$. Thus, we may replace $z$ by a point
$z^{*}$ satisfying $|z^{*}-y|=|z^{\prime}-y^{\prime}|$, or the property that
$z^{*},q,x$ are collinear. Repeating this transformation by $x$ or $y$ playing
the role of $z$ we obtain either a triangle congruent to $T^{\prime}$ in which
$q$ satisfies the required conditions, or a triangle in which $q$ is a
boundary point. In other words, without loss of generality we may assume that
$q\in\operatorname{bd}(T)$. If $q\in\\{x,y,z\\}$, then the statement is
trivial, and so we assume that $q$ is a relative interior point of, say,
$[x,y]$. In this case, if $|z-x|>|z^{\prime}-x^{\prime}|$ or
$|z-y|>|z^{\prime}-y^{\prime}|$, then we may rotate $[y,z]$ or $[x,z]$ around
$y$ or $x$, respectively. Finally, if $|y-x|>|y^{\prime}-x^{\prime}|$, then
one of the angles $\angle yxz$ or $\angle xyz$, say $\angle xyz$, is acute,
and then we may rotate $[z,y]$ around $z$ towards $[z,q]$. This implies the
statement.

By our argument, it is sufficient to prove Lemma 5 under the assumption that
$|y-x|=\rho_{x}+{\nu}\rho_{y}$, $|z-x|=\rho_{x}+{\nu}\rho_{z}$ and
$|z-y|=\rho_{y}+{\nu}\rho_{z}$. Consider the case that $\rho_{x}>\rho_{y}$.
Let $q$ be a point of $T$ belonging to each disk, implying that
$|q-u|\leq\rho_{u}$ for all $u\in\\{x,y,z\\}$. Clearly, from our conditions it
follows that $|x-q|>\rho_{x}-\rho_{y}$. Let us define a $1$-parameter family
of configurations, with the parameter $t\in[0,\rho_{x}-\rho_{y}]$, by setting
$x(t)=x-tw$, where $w$ is the unit vector in the direction of $x-q$,
$\rho_{x}(t)=\rho_{x}-t$, and keeping $q,y,z,\rho_{y},\rho_{z}$ fixed. Note
that in this family $q\in B_{x(t)}=x(t)+\rho_{x}(t)\mathbf{B}^{2}$, which
implies that $|x(t)-u|\leq\rho_{x}(t)+\rho_{u}$ for $u\in\\{y,z\\}$. Thus, for
any $\\{u,v\\}\subset\\{x(t),y,z\\}$, there is a point of $[u,v]$ belonging to
both $B_{u}$ and $B_{v}$. This, together with the property that $q$ belongs to
all three disks and using the convexity of the disks, yields that the triangle
$[x(t),y,z]$ is covered by $B_{x(t)}\cup B_{y}\cup B_{z}$.

Let the angle between $u-x(t)$ and $w$ be denoted by $\varphi$. Then, using
the linearity of directional derivatives, we have that for $f(t)=|x(t)-u|$,
$f^{\prime}(t)=-\cos\varphi\geq-1$ for $u\in\\{y,z\\}$, implying
$|x(t)-u|\geq|x-u|-t=\rho_{x}(t)+\nu\rho_{u}$ for $u\in\\{y,z\\}$, and also
that the configuration is a $\nu$-arrangement for all values of $t$. Hence,
all configurations in this family, and in particular, the configuration with
$t=\rho_{x}-\rho_{y}$ satisfies the conditions in the lemma. Thus, repeating
again the argument in the first part of the proof, we may assume that
$\rho_{x}=\rho_{y}\geq\rho_{z}$, $|y-x|=(1+\mu)\rho_{x}$ and
$|z-x|=|z-y|=\rho_{x}+{\nu}\rho_{z}$. Finally, if
$\rho_{x}=\rho_{y}>\rho_{z}$, then we may assume that $q$ lies on the symmetry
axis of $T$ and satisfies $|x-q|=|y-q|>\rho_{x}-\rho_{z}$. In this case we
apply a similar argument by moving $x$ and $y$ towards $q$ at unit speed and
decreasing $\rho_{x}=\rho_{y}$ simultaneously till they reach $\rho_{z}$, and,
again repeating the argument in the first part of the proof, obtain that the
family
$\\{\bar{u}+\rho_{z}\mathbf{B}^{2}:\bar{u}\in\\{\bar{x},\bar{y},\bar{z}\\}\\}$,
where $\bar{T}=[\bar{x},\bar{y},\bar{z}]$ is a regular triangle of side
lengths $(1+\nu)\rho_{z}$, covers $\bar{T}$. Thus, the inequality
$\nu\leq\sqrt{3}-1$ follows by an elementary computation. ∎

In our next lemma, for any disk $B_{i}\in\mathcal{F}$ we denote by
$\bar{B}_{i}$ the closure $x_{i}+\rho_{i}\mathbf{B}^{2}$ of $B_{i}$.

######  Lemma 6.

Let $B_{i},B_{j},B_{k}\in\mathcal{F}$ such that
$\bar{B}_{u}\cap\bar{B}_{v}\not\subseteq B_{w}$ for any
$\\{u,v,w\\}=\\{i,j,k\\}$. Let $T=[x_{i},x_{j},x_{k}]$,
$\Delta=\operatorname{area}(T)$, and $\alpha_{u}=\angle x_{v}x_{u}x_{w}$. If
$T\subset{\bar{B}_{i}\cup\bar{B}_{j}\cup\bar{B}_{k}}$, then

(3)
$\frac{1}{2}\sum_{u\in\\{i,j,k\\}}\alpha_{u}\rho_{u}^{2}\leq\frac{2\pi}{\sqrt{3}(1+\mu)^{2}}\Delta,$

with equality if and only if $\rho_{i}=\rho_{j}=\rho_{k}$, and $T$ is a
regular triangle of side length $(1+\mu)\rho_{i}$.

###### Proof.

In the proof we call

$\delta=\frac{\sum_{u\in\\{i,j,k\\}}\alpha_{u}\rho_{u}^{2}}{2\Delta}$

the _density_ of the configuration.

Consider the $1$-parameter families of disks
$B_{u}(\nu)=x_{u}+\left(1+\mu\right)/\left({1+\nu}\right)\rho_{u}\operatorname{int}(\mathbf{B}^{2})$,
where $u\in\\{i,j,k\\}$ and $\nu\in[\mu,1]$. Observe that the three disks
$B_{u}(\nu)$, where $u\in\\{i,j,k\\}$, form a $\nu$-arrangement for any
$\nu\geq\mu$. Indeed, in this case for any $\\{u,v\\}\subset\\{i,j,k\\}$, if
$\rho_{u}\leq\rho_{v}$, we have

$\frac{1+\mu}{1+\nu}\rho_{v}+\nu\left(\frac{1+\mu}{1+\nu}\rho_{u}\right)=\rho_{v}+\mu\rho_{u}-\frac{\nu-\mu}{1+\nu}(\rho_{v}-\rho_{u})\leq\rho_{v}+\mu\rho_{u}\leq|x_{u}-x_{v}|.$

Furthermore, for any $\nu\geq\mu$, we have

$(1+\mu)^{2}\sum_{u\in\\{i,j,k\\}}\alpha_{u}\rho_{u}^{2}=(1+\nu)^{2}\sum_{u\in\\{i,j,k\\}}\alpha_{u}\left(\frac{1+\mu}{1+\nu}\right)^{2}\rho_{u}^{2}.$

Thus, it is sufficient to prove the assertion for the maximal value
$\bar{\nu}$ of $\nu$ such that the conditions
$T\subset\bar{B}_{i}(\nu)\cup\bar{B}_{j}(\nu)\cup\bar{B}_{k}(\nu)$ and
$\bar{B}_{u}\cap\bar{B}_{v}\not\subseteq B_{w}$ are satisfied for any
$\\{u,v,w\\}=\\{i,j,k\\}$. Since the relation
$\bar{B}_{u}\cap\bar{B}_{v}\not\subseteq B_{w}$ implies, in particular, that
$\bar{B}_{u}\cap\bar{B}_{v}\neq\emptyset$, in this case the conditions of
Lemma 5 are satisfied, yielding $\bar{\nu}\leq\sqrt{3}-1$. Hence, with a
little abuse of notation, we may assume that $\bar{\nu}=\mu$. Then one of the
following holds:

  1. (i)

The intersection of the disks $\bar{B}_{u}$ is a single point.

  2. (ii)

For some $\\{u,v,w\\}=\\{i,j,k\\}$,
$\bar{B}_{u}\cap\bar{B}_{v}\subset\bar{B}_{w}$ and
$\bar{B}_{u}\cap\bar{B}_{v}\not\subset B_{w}$.

Before investigating (i) and (ii), we remark that during this process, which
we refer to as _$\mu$ -increasing process_, even though there might be non-
maximal values of $\nu$ for which the modified configuration satisfies the
conditions of the lemma and also (i) or (ii), we always choose the _maximal_
value. This value is determined by the centers of the original disks and the
ratios of their radii.

First, consider (i). Then, clearly, the unique intersection point $q$ of the
disks lies in $T$, and note that either $q$ lies in the boundary of all three
disks, or two disks touch at $q$. We describe the proof only in the first
case, as in the second one we may apply a straightforward modification of our
argument. Thus, in this case we may decompose $T$ into three triangles
$[x_{i},x_{j},q]$, $[x_{i},x_{k},q]$ and $[x_{j},x_{k},q]$ satisfying the
conditions in Lemma 4, and obtain

$\frac{1}{2}\sum_{u\in\\{i,j,k\\}}\alpha_{u}\rho_{u}^{2}\leq\frac{4\arccos\frac{1+\mu}{2}}{(1+\mu)\sqrt{(1-\mu)(3+\mu)}}\Delta\leq\frac{2\pi}{\sqrt{3}(1+\mu)^{2}}\Delta,$

where the second inequality follows from the fact that the two expressions are
equal if $\mu=\sqrt{3}-1$, and

$\left(2\arccos\frac{1+\mu}{2}-\frac{\pi\sqrt{(1-\mu)(3+\mu)}}{\sqrt{3}(1+\mu)}\right)^{\prime}>0$

if $\mu\in[0,\sqrt{3}-1]$. Here, by Lemma 4, equality holds only if
$\rho_{i}=\rho_{j}=\rho_{k}$, and $T$ is a regular triangle of side length
$(1+\mu)\rho_{i}$. On the other hand, under these conditions in (3) we have
equality. This implies Lemma 6 for (i).

In the remaining part of the proof, we show that if (ii) is satisfied, the
density of the configuration is strictly less than
${2\pi}/\left(\sqrt{3}(1+\mu)^{2}\right)$. Let $q$ be a common point of
$\operatorname{bd}(\bar{B}_{w})$ and, say, $\bar{B}_{u}$. If $q$ is a relative
interior point of an arc in $\operatorname{bd}(\bar{B}_{u}\cap\bar{B}_{v})$,
then one of the disks is contained in another one, which contradicts the fact
that the disks $B_{u},B_{v},B_{w}$ form a $\mu$-arrangement. Thus, we have
that either $\bar{B}_{u}\cap\bar{B}_{v}=\\{q\\}$, or that $q$ is a vertex of
the digon $B_{u}\cap B_{v}$. If $\bar{B}_{u}\cap\bar{B}_{v}=\\{q\\}$, then the
conditions of (i) are satisfied, and thus, we assume that $q$ is a vertex of
the digon $B_{u}\cap B_{v}$. By choosing a suitable coordinate system and
rescaling and relabeling, if necessary, we may assume that
$B_{u}=\operatorname{int}(\mathbf{B}^{2})$, $x_{v}$ lies on the positive half
of the $x$-axis, and $x_{w}$ is written in the form
$x_{w}=(\zeta_{w},\eta_{w})$, where $\eta_{w}>0$, and the radical line of
$B_{u}$ and $B_{v}$ separates $x_{v}$ and $x_{w}$ (cf. Figure 5). Set
$\rho=\rho_{w}$. We show that $\eta_{w}>(1+\mu)\rho/{2}$.

\begin{overpic}[width=303.53267pt]{jouvw.pdf} \put(30.0,23.0){$\bar{B}_{u}$}
\put(43.0,33.0){$\bar{B}_{w}$} \put(60.0,23.0){$\bar{B}_{v}$} \end{overpic}
Figure 5. An illustration for the proof of Lemma 6.

_Case 1_ , if $\rho\geq 1$. Then we have $|x_{w}|\geq\rho+\mu$.

Let the radical line of $B_{u}$ and $B_{v}$ be the line $\\{x=t\\}$ for some
$0<t\leq 1$. Then, as this line separates $x_{v}$ and $x_{w}$, we have
$\zeta_{w}\leq t$, and by (ii) we have $q=(t,-\sqrt{1-t^{2}})$. This implies
that $|x_{w}-q|\leq|x_{w}-x_{u}|,|x_{w}-x_{v}|$, from which we have
$0\leq\zeta_{w}$. Let $S$ denote the half-infinite strip
$S=\\{(\zeta,\eta)\in\mathbb{R}^{2}:0\leq\zeta\leq t,\eta\geq 0\\}$, and set
$s=(t,-\sqrt{1-t^{2}}+\rho)$. Note that by our considerations, $x_{w}\in S$
and $|x_{w}-q|=\rho$, which yield $\eta_{w}\leq-\sqrt{1-t^{2}}+\rho$. From
this it follows that $\rho+\mu\leq|x_{w}|\leq|s|$, or in other words, we have
$t^{2}+(\rho-\sqrt{1-t^{2}})^{2}\geq(\rho+\mu)^{2}$. By solving this
inequality for $t$ with parameters $\rho$ and $\mu$, we obtain that $t\geq
t_{0}$, $1\leq\rho\leq\left(1-\mu^{2}\right)/\left(2\mu\right)$ and
$0\leq\mu\leq\sqrt{2}-1$, where

$t_{0}=\sqrt{1-\left(\frac{1-2\mu\rho-\mu^{2}}{2\rho}\right)^{2}}.$

Let $p=(\zeta_{p},\eta_{p})$ be the unique point in $S$ with $|p|=\rho+\mu$
and $|p-q|=\rho$, and observe that $\eta_{w}\geq\eta_{p}$. Now we find the
minimal value of $\eta_{p}$ if $t$ is permitted to change and $\rho$ is fixed.
Set $p^{\prime}=(\zeta_{p},-\sqrt{1-\zeta_{p}^{2}})$. Since the bisector of
$[p^{\prime},q]$ separates $p^{\prime}$ and $p$, it follows that
$|p-p^{\prime}|\geq|p-q|=\rho$ with equality only if $p^{\prime}=q$ and $p=s$,
or in other words, if $t=t_{0}$. This yields that $\zeta_{p}$ is maximal if
$t=t_{0}$. On the other hand, since $|p|=\rho+\mu$ and $p$ lies in the first
quadrant, $\eta_{p}$ is minimal if $\zeta_{p}$ is maximal. Thus, for a fixed
value of $\rho$, $\eta_{p}$ is minimal if $t=t_{0}$ and
$p=s=(t_{0},-\sqrt{1-t_{0}^{2}}+\rho)$, implying that
$\eta_{w}\geq-\sqrt{1-t_{0}^{2}}+\rho=\left(2\rho^{2}+\mu^{2}+2\mu\rho-1\right)/(2\rho)$.
Now, $\rho\geq 1$ and $\mu<1$ yields that

$\frac{2\rho^{2}+\mu^{2}+2\mu\rho-1}{2\rho}-\frac{(1+\mu)\rho}{2}=\frac{\rho^{2}-\mu\rho^{2}+2\mu\rho-1}{2\rho}\geq\frac{\mu}{2\rho}>0,$

implying the statement.

_Case 2_ , if $0<\rho\leq 1$. In this case the inequality
$\eta_{w}>{(1+\mu)\rho}/{2}$ follows by a similar consideration.

In the remaining part of the proof, let

$\sigma(\mu)=\frac{2\pi}{\sqrt{3}(1+\mu)^{2}}.$

Now we prove the lemma for (ii). Suppose for contradiction that for some
configuration $\\{B_{u},B_{v},B_{w}\\}$ satisfying (ii) the density is at
least $\sigma(\mu)$; here we label the disks as in the previous part of the
proof. Let
$B^{\prime}_{w}=x^{\prime}_{w}+\rho_{w}\operatorname{int}(\mathbf{B}^{2})$
denote the reflection of $B_{w}$ to the line through $[x_{u},x_{v}]$. By the
inequality $\eta_{w}>{(1+\mu)\rho}/{2}$ proved in the two previous cases, we
have that $\\{B_{u},B_{v},B_{w},B_{w}^{\prime}\\}$ is a $\mu$-arrangement,
where we observe that by the strict inequality, $B_{w}$ and $B_{w}^{\prime}$
do not touch each others cores. Furthermore, each triangle
$[x_{u},x_{w},x^{\prime}_{w}]$ and $[x_{v},x_{w},x^{\prime}_{w}]$ is covered
by the three disks from this family centered at the vertices of the triangle,
and the intersection of no two disks from one of these triples is contained in
the third one. Thus, the conditions of Lemma 6 are satisfied for both
$\\{B_{u},B_{w},B^{\prime}_{w}\\}$ and $\\{B_{v},B_{w},B^{\prime}_{w}\\}$.
Observe that as by our assumption the density in $T$ is $\sigma(\mu)$, it
follows that the density in at least one of the triangles
$[x_{u},x_{w},x^{\prime}_{w}]$ and $[x_{v},x_{w},x^{\prime}_{w}]$, say in
$T^{\prime}=[x_{u},x_{w},x^{\prime}_{w}]$, is at least $\sigma(\mu)$. In other
words, under our condition there is an axially symmetric arrangement with
density at least $\sigma(\mu)$. Now we apply the $\mu$-increasing process as
in the first part of the proof and obtain a $\mu^{\prime}$-arrangement
$\\{\hat{B}_{u}=x_{u}+{(1+\mu)}/{(1+\mu^{\prime})}\rho_{u}\operatorname{int}(\mathbf{B}^{2}),\hat{B}_{w}=x_{w}+{(1+\mu)}/{(1+\mu^{\prime})}\rho_{w}\operatorname{int}(\mathbf{B}^{2}),\hat{B}_{w}^{\prime}=x_{w}^{\prime}+{(1+\mu)}/{(1+\mu^{\prime})}\rho_{w}\operatorname{int}(\mathbf{B}^{2})\\}$
with density ${\sigma(\mu^{\prime})}$ and $\mu^{\prime}\geq\mu$ that satisfies
either (i) or (ii). If it satisfies (i), we have that the density of this
configuration is at most $\sigma(\mu^{\prime})$ with equality if only if
$T^{\prime}$ is a regular triangle of side length $(1+\mu^{\prime})\rho$,
where $\rho$ is the common radius of the three disks. On the other hand, this
implies that in case of equality, the disks centered at $x_{w}$ and
$x_{w}^{\prime}$ touch each others’ cores which, by the properties of the
$\mu$-increasing process, contradicts the fact that $B_{w}$ and
$B_{w}^{\prime}$ do not touch each others’ $\mu$-cores. Thus, we have that the
configuration satisfies (ii).

From Lemma 1 it follows that
$\hat{B}_{w}\cap\hat{B}_{w}^{\prime}\subset\hat{B}_{u}$. Thus, applying the
previous consideration with $\hat{B}_{u}$ playing the role of $B_{w}$, we
obtain that the distance of $x_{u}$ from the line through
$[x_{w},x_{w}^{\prime}]$ is greater than ${(1+\mu^{\prime})}/{2}\rho_{u}$.
Thus, defining
$\hat{B}_{u}^{\prime}=x_{u}^{\prime}+{(1+\mu)}/{(1+\mu^{\prime})}\rho_{u}\operatorname{int}(\mathbf{B}^{2})$
as the reflection of $B_{u}^{\prime}$ about the line through
$[x_{w},x_{w}^{\prime}]$, we have that
$\\{\hat{B}_{u},\hat{B}_{w},\hat{B}_{w}^{\prime},\hat{B}_{u}^{\prime}\\}$ is a
$\mu^{\prime}$-arrangement such that
$\\{\hat{B}_{u},\hat{B}_{u}^{\prime},\hat{B}_{w}\\}$ and
$\\{\hat{B}_{u},\hat{B}_{u}^{\prime},\hat{B}_{w}^{\prime}\\}$ satisfy the
conditions of Lemma 6. Without loss of generality, we may assume that the
density of $\\{\hat{B}_{u},\hat{B}_{u}^{\prime},\hat{B}_{w}\\}$ is at least
$\sigma(\mu^{\prime})$. Again applying the $\mu$-increasing procedure
described in the beginning of the proof, we obtain a
$\mu^{\prime\prime}$-arrangement of three disks, with
$\mu^{\prime\prime}\geq\mu^{\prime}$, concentric with the original ones that
satisfy the conditions of the lemma and also (i) or (ii). Like in the previous
paragraph, (i) leads to a contradiction, and we have that it satisfies (ii).
Now, again repeating the argument we obtain a $\mu^{\prime\prime\prime}$
-arrangement

$\left\\{y+\frac{1+\mu}{1+\mu^{\prime\prime\prime}}\rho_{u}\operatorname{int}(\mathbf{B}^{2}),x_{w}+\frac{1+\mu}{1+\mu^{\prime\prime\prime\prime}}\rho_{w}\operatorname{int}(\mathbf{B}^{2}),x_{w}^{\prime}+\frac{1+\mu}{1+\mu^{\prime\prime\prime}}\rho_{w}\operatorname{int}(\mathbf{B}^{2})\right\\},$

with density at least $\sigma(\mu^{\prime\prime\prime})$ and
$\mu^{\prime\prime\prime}\geq\mu^{\prime\prime}$, that satisfies the
conditions of the lemma, where either $y=x_{u}$ or $y=x^{\prime}_{u}$. On the
other hand, since in the $\mu$-increasing process we choose the _maximal_
value of the parameter satisfying the required conditions, this yields that
$\mu^{\prime}=\mu^{\prime\prime}=\mu^{\prime\prime\prime}$. But in this case
the property that $\\{\hat{B}_{u},\hat{B}_{u}^{\prime},\hat{B}_{w}\\}$
satisfies (ii) yields that
$\\{\hat{B}_{u},\hat{B}_{u}^{\prime},\hat{B}_{w}\\}$ does not; a
contradiction. ∎

##  3\. Proof of Theorem 1

The idea of the proof follows that in [9] with suitable modifications. In the
proof we decompose $U(\mathcal{F})=\bigcup_{i=1}^{n}B_{i}$, by associating a
polygon to each vertex of certain free digons formed by two disks. Before
doing it, we first prove some properties of $\mu$-arrangements.

Let $q$ be a vertex of a free digon, say, $D=B_{1}\cap B_{2}$. We show that
the convex angular region $R$ bounded by the closed half lines starting at $q$
and passing through $x_{1}$ and $x_{2}$, respectively, does not contain the
center of any element of $\mathcal{F}$ different from $B_{1}$ and $B_{2}$
containing $q$ on its boundary. Indeed, suppose for contradiction that there
is a disk
$B_{3}=x_{3}+\rho_{3}\operatorname{int}(\mathbf{B}^{2})\in\mathcal{F}$ with
$q\in{\operatorname{bd}(B_{3})}$ and $x_{3}\in R$. Since
$[q,x_{1},x_{2}]{\setminus\\{q\\}}\subset B_{1}\cup B_{2}$, from this and the
fact that $\mathcal{F}$ is a Minkowski-arrangement, it follows that the line
through $[x_{1},x_{2}]$ strictly separates $x_{3}$ from $q$. As this line is
the bisector of the segment $[q,q^{\prime}]$, where $q^{\prime}$ is the vertex
of $D$ different from $q$, from this it also follows that
$|x_{3}-q|>|x_{3}-q^{\prime}|$. Thus, $q^{\prime}\in B_{3}$.

Observe that in a Minkowski arrangement any disk intersects the boundary of
another one in an arc shorter than a semicircle. This implies, in particular,
that $B_{3}\cap\operatorname{bd}(B_{1})$ and
$B_{3}\cap\operatorname{bd}(B_{2})$ are arcs shorter than a semicircle. On the
other hand, from this the fact that $q,q^{\prime}\in B_{3}$ yields that
$\operatorname{bd}(D)\subset B_{3}$, implying, by the properties of convexity,
that $D\subset B_{3}$, which contradicts our assumption that $D$ is a free
digon.

Note that, in particular, we have shown that if a member of $\mathcal{F}$
contains both vertices of a digon, then it contains the digon.

\begin{overpic}[width=216.81pt]{insc2.pdf} \put(22.0,28.0){$B_{1}$}
\put(65.0,28.0){$B_{2}$} \put(37.0,34.0){$B(t)$} \end{overpic} Figure 6. The
$1$-parameter family of disks inscribed in $B_{1}\cap B_{2}$.

Observe that the disks inscribed in $D$ can be written as a $1$-parameter
family of disks $B(t)$ continuous with respect to Hausdorff distance, where
$t\in(0,1)$ and $B(t)$ tends to $\\{q\\}$ as $t\to 0^{+}$ (cf. Figure 6); here
the term ‘inscribed’ means that the disk is contained in $B_{i}\cap B_{j}$ and
touches both disks from inside. We show that if some member $B_{k}$ of
$\mathcal{F}$, different from $B_{1}$ and $B_{2}$, contains $B(t)$ for some
value of $t$, then $B_{k}$ contains exactly one vertex of $D$. Indeed, assume
that some $B_{k}$ contains some $B(t)$ but it does not contain any vertex of
$D$. Then for $i\in\\{1,2\\}$, $B_{k}\cap\operatorname{bd}(B_{i})$ is a
circular arc $\Gamma_{i}$ in $\operatorname{bd}(D)$. Let $L_{i}$ be the half
line starting at the midpoint of $\Gamma_{i}$, and pointing in the direction
of the outer normal vector of $B_{i}$ at this point. Note that as $D$ is a
plane convex body, $L_{1}\cap L_{2}=\emptyset$. On the other hand, since
$B_{1},B_{2},B_{k}$ are a Minkowski arrangement, from this it follows that
$x_{k}\in L_{1}\cap L_{2}$; a contradiction. The property that no $B_{k}$
contains both vertices of $D$ follows from the fact that $D$ is a free digon.
Thus, if $q\in B_{k}$ for an element $B_{k}\in\mathcal{F}$, then there is some
value $t_{0}\in{(0,1)}$ such that $B(t)\subseteq B_{k}$ if and only if
$t\in(0,t_{0}]$.

In the proof, we call the disks $B_{i},B_{j}$ _adjacent_ , if $B_{i}\cap
B_{j}$ is a digon, and there is a member of the family $B(t)$ defined in the
previous paragraph that is not contained in any element of $\mathcal{F}$
different from $B_{i}$ and $B_{j}$. Here, we remark that any two adjacent
disks define a free digon, and if a vertex of a free digon is a boundary point
of $U(\mathcal{F})$, then the digon is defined by a pair of adjacent disks.

Consider a pair of adjacent disks, say $B_{1}$ and $B_{2}$, and let $q$ be a
vertex of $D=B_{1}\cap B_{2}$. If $q$ is a boundary point of the union
$U(\mathcal{F})$, then we call the triangle $[x_{1},x_{2},q]$ a _shell
triangle_ , and observe that by the consideration in the previous paragraph,
the union of shell triangles coincides with the inner shell of $\mathcal{F}$.

If $q$ is not a boundary point of $U(\mathcal{F})$, then there is a maximal
value $t_{0}\in{(0,1)}$ such that $B(t_{0})=x+\rho\mathbf{B}^{2}$ is contained
in an element $B_{i}$ of $\mathcal{F}$ satisfying $q\in B_{i}$. Then, clearly,
$B(t_{0})$ touches any such $B_{i}$ from inside, and since $B_{1}$ and $B_{2}$
are adjacent, there is no element of $\mathcal{F}$ containing $B(t_{0})$ and
the vertex of $D$ different from $q$. Without loss of generality, assume that
the elements of $\mathcal{F}$ touched by $B(t_{0})$ from inside are
$B_{1},B_{2},\ldots,B_{k}$. Since $B_{1}$ and $B_{2}$ are adjacent and there
is no element of $\mathcal{F}$ containing both $B(t_{0})$ and the vertex of
$D$ different from $q$, we have that the tangent points of $B_{1}$ and $B_{2}$
on $\operatorname{bd}(B(t_{0}))$ are consecutive points among the tangent
points of all the disks $B_{i}$, where $1\leq i\leq k$. Thus, we may assume
that the tangent points of $B_{1},B_{2},\ldots,B_{k}$ on $B(t_{0})$ are in
this counterclockwise order on $\operatorname{bd}(B(t_{0}))$. Let $x$ denote
the center of $B(t_{0})$. Since $\mathcal{F}$ is a Minkowski arrangement, for
any $1\leq i<j\leq k$, the triangle $[x,x_{i},x_{j}]$ contains the center of
no element of $\mathcal{F}$ apart from $B_{i}$ and $B_{j}$, which yields that
the points $x_{1},x_{2},\ldots,x_{k}$ are in convex position, and their convex
hull $P_{q}$ contains $x$ in its interior but it does not contain the center
of any element of $\mathcal{F}$ different from $x_{1},x_{2},\ldots,x_{k}$ (cf.
also [9]). We call $P_{q}$ a _core polygon_.

We remark that since $\mathcal{F}$ is a $\mu$-arrangement, the longest side of
the triangle $[x,x_{i},x_{i+1}]$, for $i=1,2\ldots,k$, is $[x_{i},x_{i+1}]$.
This implies that $\angle x_{i}xx_{i+1}>{\pi}/{3}$, and also that $k<6$.
Furthermore, it is easy to see that for any $i=1,2,\ldots,k$, the disks
$B_{i}$ and $B_{i+1}$ are adjacent. Thus, any edge of a core polygon is an
edge of another core polygon or a shell triangle. This property, combined with
the observation that no core polygon or shell triangle contains any center of
an element of $\mathcal{F}$ other than their vertices, implies that core
polygons cover the core of $\mathcal{F}$ without interstices and overlap (see
also [9]).

Let us decompose all core polygons of $\mathcal{F}$ into triangles, which we
call _core triangles_ , by drawing all diagonals in the polygon starting at a
fixed vertex, and note that the conditions in Lemma 6 are satisfied for all
core triangles. Now, the inequality part of Theorem 1 follows from Lemmas 4
and 6, with equality if and only if each core triangle is a regular triangle
$[x_{i},x_{j},x_{k}]$ of side length $(1+\mu)\rho$, where
$\rho=\rho_{i}=\rho_{j}=\rho_{k}$, and each shell triangle $[x_{i},x_{j},q]$,
where $q$ is a vertex of the digon $B_{i}\cap B_{j}$ is an isosceles triangle
whose base is of length $(1+\mu)\rho$, and $\rho=\rho_{i}=\rho_{j}$.
Furthermore, since to decompose a core polygon into core triangles we can draw
diagonals starting at any vertex of the polygon, we have that in case of
equality in the inequality in Theorem 1, all sides and all diagonals of any
core polygon are of equal length. From this we have that all core polygons are
regular triangles, implying that all free digons in $\mathcal{F}$ are thick.

On the other hand, assume that all free digons in $\mathcal{F}$ are thick.
Then, from Lemma 2 it follows that any connected component of $\mathcal{F}$
contains congruent disks. Since an adjacent pair of disks defines a free
digon, from this we have that, in a component consisting of disks of radius
$\rho>0$, the distance between the centers of two disks defining a shell
triangle, and the edge-lengths of any core polygon, are equal to
$(1+\mu)\rho$. Furthermore, since all disks centered at the vertices of a core
polygon are touched by the same disk from inside, we also have that all core
polygons in the component are regular $k$-gons of edge-length $(1+\mu)\rho$,
where $3\leq k\leq 5$. This and the fact that any edge of a core polygon
connects the vertices of an adjacent pair of disks yield that if the
intersection of any two disks centered at two different vertices of a core
polygon is more than one point, then it is a free digon. Thus, any diagonal of
a core polygon in this component is of length $(1+\mu)\rho$, implying that any
core polygon is a regular triangle, from which the equality in Theorem 1
readily follows.

##  4\. Remarks and open questions

######  Remark 1.

If $\sqrt{3}-1<\mu<1$, then by Lemma 5, $C(\mathcal{F})=\emptyset$ for any
$\mu$-arrangement $\mathcal{F}$ of order $\mu$.

######  Remark 2.

Observe that the proof of Theorem 1 can be extended to some value
$\mu>\sqrt{3}-1$ if and only if Lemma 4 can be extended to this value $\mu$.
Nevertheless, from the continuity of the functions in the proof of Lemma 4, it
follows that there is some $\mu_{0}>\sqrt{3}-1$ such that the lemma holds for
any $\mu\in(\sqrt{3}-1,\mu_{0}]$. Nevertheless, we cannot extend the proof for
all $\mu<1$ due to numeric problems.

Remark 2 readily implies Remark 3.

######  Remark 3.

There is some $\mu_{0}>\sqrt{3}-1$ such that if $\mu\in(\sqrt{3}-1,\mu_{0}]$,
and $\mathcal{F}$ is a $\mu$-arrangment of finitely many disks, then the total
area of the disks is

$T\leq\frac{4\cdot\arccos({\frac{1+\mu}{2})}}{(1+\mu)\cdot\sqrt{(3+\mu)(1-\mu)}}\operatorname{area}(I(\mathcal{F}))+\operatorname{area}(O(\mathcal{F})),$

with equality if and only if every free digon in $\mathcal{F}$ is thick.

######  Conjecture 1.

The statement in Remark 3 holds for any $\mu$-arrangement of finitely many
disks with $\sqrt{3}-1<\mu<1$.

Let $0<\mu<1$ and let $\mathcal{F}=\\{K_{i}:i=1,2,\ldots\\}$ be a generalized
Minkowski arrangement of order $\mu$ of homothets of an origin-symmetric
convex body in $\mathbb{R}^{d}$ with positive homogeneity. Then we define the
(upper) density of $\mathcal{F}$ with respect to $U(\mathcal{F})$ as

$\delta_{U}(\mathcal{F})=\limsup_{R\to\infty}\frac{\sum_{B_{i}\subset
R\mathbf{B}^{2}}\operatorname{area}\left(B_{i}\right)}{\operatorname{area}\left(\bigcup_{B_{i}\subset
R\mathbf{B}^{2}}B_{i}\right)}.$

Clearly, we have $\delta(\mathcal{F})\leq\delta_{U}(\mathcal{F})$ for any
arrangement $\mathcal{F}$.

Our next statement is an immediate consequence of Theorem 1 and Remark 3.

######  Corollary 1.

There is some value $\sqrt{3}-1<\mu_{0}<1$ such that for any $\mu$-arrangement
$\mathcal{F}$ of Euclidean disks in $\mathbb{R}^{2}$, we have

$\delta_{U}(\mathcal{F})\leq\left\\{\begin{array}[]{l}\frac{2\pi}{\sqrt{3}(1+\mu)^{2}},\hbox{
if }0\leq\mu\leq\sqrt{3}-1,\hbox{and}\\\
\frac{4\cdot\arccos({\frac{1+\mu}{2})}}{(1+\mu)\cdot\sqrt{(3+\mu)(1-\mu)}},\hbox{
if }\sqrt{3}-1<\mu\leq\mu_{0}.\end{array}\right.$

For any $0\leq\mu<1$, let $u,v\in\mathbb{R}^{2}$ be two unit vectors whose
angle is $\frac{\pi}{3}$, and let $\mathcal{F}_{hex}(\mu)$ denote the family
of disks of radius $(1+\mu)$ whose set of centers is the lattice
$\\{ku+mv:k,m\in\mathbb{Z}\\}$. Then $\mathcal{F}_{hex}(\mu)$ is a
$\mu$-arrangement, and by Corollary 1, for any $\mu\in[0,\sqrt{3}-1]$, it has
maximal density on the family of $\mu$-arrangements of positive homogeneity.
Nevertheless, as Fejes Tóth observed in [8] (see also [5] or Section 1), the
same does not hold if $\mu>\sqrt{3}-1$. Indeed, an elementary computation
shows that in this case $\mathcal{F}_{hex}(\mu)$ does not cover the plane, and
thus, by adding disks to it that lie in the uncovered part of the plane we can
obtain a $\mu$-arrangement with greater density.

Fejes Tóth suggested the following construction to obtain $\mu$-arrangements
with large densities. Let $\tau>0$ be sufficiently small, and, with a little
abuse of notation, let $\tau\mathcal{F}_{\operatorname{hex}}(\mu)$ denote the
family of the homothetic copies of the disks in
$\mathcal{F}_{\operatorname{hex}}(\mu)$ of homothety ratio $\tau$ and the
origin as the center of homothety. Let
$\mathcal{F}^{1}_{\operatorname{hex}}(\mu)$ denote the $\mu$-arrangement
obtained by adding those elements of
$\tau\mathcal{F}_{\operatorname{hex}}(\mu)$ to
$\mathcal{F}_{\operatorname{hex}}(\mu)$ that do not overlap any element of it.
Iteratively, if for some positive integer $k$,
$\mathcal{F}^{k}_{\operatorname{hex}}(\mu)$ is defined, then let
$\mathcal{F}^{k+1}_{\operatorname{hex}}(\mu)$ denote the union of
$\mathcal{F}^{k}_{\operatorname{hex}}(\mu)$ and the subfamily of those
elements of $\tau^{k+1}\mathcal{F}_{\operatorname{hex}}(\mu)$ that do not
overlap any element of it. Then, as was observed also in [8], choosing
suitable values for $\tau$ and $k$, the value of
$\delta_{U}(\mathcal{F}_{\operatorname{hex}}(\mu))$ can be approximated
arbitrarily well by $\delta(\mathcal{F}^{k}_{\operatorname{hex}}(\mu))$. We
note that the same idea immediately leads to the following observation.

######  Remark 4.

The supremums of $\delta(\mathcal{F})$ and $\delta_{U}(\mathcal{F})$ coincide
on the family of the $\mu$-arrangements $\mathcal{F}$ in $\mathbb{R}^{2}$ of
positive homogeneity.

We finish the paper with the following conjecture.

######  Conjecture 2.

For any $\mu\in(\sqrt{3}-1,1)$ and any $\mu$-arrangement $\mathcal{F}$ in
$\mathbb{R}^{2}$, we have
$\delta(\mathcal{F})\leq\delta_{U}(\mathcal{F}_{\operatorname{hex}}(\mu))$.

Acknowledgments.
The authors express their gratitude to K. Bezdek for directing their attention
to this interesting problem, and to two anonymous referees for many helpful
suggestions.

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Uncertainty in Climate Science: Not Cause for Inaction111This article appeared
in print in two parts (without the material in the appendix of the present
article), covering separately the two themes of this article: J.M. Restrepo,
M. Mann, This is how Climate is Always Changing, SIAM News May 2018, and J.M.
Restrepo, M. Mann, This is how Climate is Always Changing, Focus Group on
Climate Newsletter, American Physical Society, March 2018. For citation
purposes please use both of these articles. If a technical aspect presented in
the appendix of this paper are required, please add a citation to this article
as well.

Juan M. Restrepo,

Department of Mathematics, and College of Earth, Oceans and Atmospheric
Sciences, Oregon State University, Corvallis OR 97331,

Michael E. Mann,

Department of Meteorology & Atmospheric Science, and The Earth and
Environmental Systems Institute, The Pennsylvania State University University
Park, PA 16802

The Fourth National Assessment, Climate Science Special Report of the US
Global Change Program, published in November 2017, concludes, “based on
extensive evidence, that it is extremely likely that human activities,
especially emissions of greenhouse gases, are the dominant cause of the
observed warming since the mid-20th century."

When asked about some of the conclusions in the report regarding systematic
climate change, Mr. Raj Shah, a spokesman for the Trump administration,
stated, "The climate has changed and it is always changing. " Shah is echoing
assertions from other observers that there is nothing unusual about the
changes in climate and weather that we are experiencing: There have been
changes before the industrial era, and some of these have been extreme.
Translated to more technical terms, such observers claim that climate has a
stationary statistical distribution –one that does not change with time– and
that, in recent years, we just happen to be experiencing samples of this
distribution that are, possibly rare, extreme highs. Shah continues, "[As the
report] states, the magnitude of future climate change depends significantly
on remaining uncertainty in the sensitivity of Earth’s climate to greenhouse
gas emissions" [21]. This part is hard to interpret, but it is meant to imply
that climate forecasting is made unreliable by the presence of uncertainties.
Shah’s statement is fully consistent with the US Environmental Protection
Agency’s (EPA) Pruitt’s statements, such as “I think that measuring with
precision human activity on the climate is something very challenging to do
and there’s tremendous disagreement about the degree of impact," he told CNBC.
“So no, I would not agree that it’s a primary contributor to the global
warming that we see," Pruitt said. “But we don’t know that yet, we need to
continue to debate, continue the review and analysis” [5]. Acknowledging that
climate changes but that the anthropogenic contribution is, at the same time,
too difficult to estimate and too small to be of importance, is a frequently-
used assertion of the Trump administration. Rather than disentangling the
statement, we will instead address separately the issue of statistical
stationarity of climate time series, and how climate predictions are impacted
by uncertainties in natural and anthropogenic forcings.

To explore the assertion of a static climate distribution, we introduce here a
theorem that applies to record values of a series of random variables drawn
from a stationary distribution, such as the measured temperatures. Others have
used this approach more rigorously to examine trends in climate data [2, 9,
23]. Since the only requirement made in the theorem on the random variables is
that they derive from a stationary distribution, the failure of this theorem
to hold indicates that the distribution from which this data arises is not
stationary. The theorem or its application does not yield causal attributions
to its outcomes. Nevertheless, the use of such a simple test circumvents the
necessity to argue about data statistics based upon model outcomes.

We will also examine how the inclusion of historically-informed uncertainties
on natural and anthropogenic greenhouse gases (GHG) modify climate
predictions. To do this we will use a simple model that captures the essential
phenomenology of the radiation balance of more complete state-of-the-art
climate models. This energy balance model (EBM) will be used in what follows
to determine whether uncertainties in GHG emissions lead to qualitatively
different climate projections than those obtained without taking uncertainties
into account.

Before proceeding we should clarify what is meant by climate, as opposed to
weather. Climate and weather describe the same system, but the term ’climate’
refers to large spatio-temporal scales and ’weather’ to small ones. This
distinction is not just a convenience. While both describe the energetics,
mass and momentum exchanges of a rotating Earth, the scale determines the
prominence of the different phenomenology, weather being dominated by inertial
effects (advective processes, turbulence, waves, density dynamics, and
transient and sometimes unstable conditions) and climate by the
forced/dissipative effects (radiation and ocean and atmospheric transport).
Weather includes tornados, hurricanes, or extreme values of temperature or
rainfall. Examples of climate are the seasons, El Niño/Southern Oscillation,
the ice ages and Industrial Era global warming.

## Records in Time Series

We make the following assertion: climate temperatures are samples from a
stationary distribution. If so, a theorem that applies to stationary
distributions should be borne out by the data. We apply a theorem about record
highs and record lows (see [7]).

One draws a sequence of independent and identically-distributed (IID) samples
$X_{1},X_{2},\ldots$ from a stationary distribution. We denote a sample from
the sequence a record high (low) if its value is higher (or lower) than the
samples preceding it. The probability of a record high is
$P_{n}:=\mbox{Prob}[X_{n}>\max\\{X_{1},X_{2},\ldots,X_{(n-1)}\\}]$ (with the
obvious modifications for the record low). In a sample set of size $n$ any one
particular value has equal chance of being the greatest (lowest) value, thus
$P_{n}=1/n$. We denote as $\mathbb{E}(R)$ the expected number of records for a
stationary random sequence of size $n$. It is given by the harmonic series
$\mathbb{E}(R)=1+1/2+1/3+\cdots+1/n$. For large $n$,
$\mathbb{E}(R)=\gamma+\log(n)$, where $\gamma$ is the Euler constant.

The occurrence of record values in climate data has been carefully compared to
predictions for a stationary distribution (see, for example, [2], [9], [23]).
Just to convey a feeling for such analyses, we undertake here a much less
rigorous but hopefully illuminating look at some data. It should be cautioned
that the application of this theorem to real data is highly nontrivial. Hence,
in what follows, we will be using this exercise merely to give a suggestive
outcome.

If the theorem applies to climate data, we expect to wait increasingly long
times for each new record temperature value (either high or low) because the
probability declines as $1/(t-t_{0})$, where the time $t$ of each temperature
observation takes the place of the statistical index $n$, and $t_{0}$ is the
start of the particular temperature observations. (If the probability
distribution were symmetric we would also expect the rates of record highs and
lows to be similar).

Figure 1a compares the record high/lows obtained from a synthetic random time
series to the July temperatures measured at the Moscow station, from about
1880 to 2011 [16]. For the random time series, the highs and lows are
similarly spaced in time. However, for the Moscow temperature data, one sees
many lows occurring at the early times and none after about 1910. By contrast,
the record highs are more spaced out in time and continue through the
observation period shown. The data suggests that the theorem on records is not
fulfilled and that the rate at which record highs or lows occur at time t does
not follow $1/(t-t_{0})$.

Figures 1b and c plot temperature data (from [16]) from 30 locations in the
Northern Hemisphere. The locations were chosen at random but were mostly
concentrated around temperate zones, simply because these records tended to be
longer. The time series are not all the same length and some stations did not
report every year. The superposition of the data in Figure 1b would lead you
to believe that, over the course of the industrial revolution, a stationary
distribution of temperatures is not all that bad a statistical model. In that
figure we highlight seven temperature time series, chosen arbitrarily. The
records associated with these 7 data are plotted in Figure 1c. To facilitate
comparison, these seven data sets have been adjusted by subtracting the first
temperature in the set (thus the adjusted temperature of any of these time
series was 0).

(a) (b) (c)

Figure 1: (a) (left) records in a synthetic stationary distribution, (right)
records of July monthly temperatures at the Moscow station; (b) temperature
data, as a function of time, for 30, arbitrary locations in the Northern
Hemisphere; (c) Record values for seven temperature time series, highlighted
in (b). The adjusted temperature subtracts out the first temperature value in
the time series. The data is taken from GISS repository. We note that you
reach a time in each data set beyond which no new lows occur whereas new highs
continue to appear going forward in time. Temperature in degrees oC.

Adding more observations to the top set or more observations to the bottom set
does not change the impression that, with time, more high records will occur
than low records (the low records stop occurring). The key observation is that
the record highs and the record lows do not obey the $1/t$ dependence. Another
test would be to compute the expectation of the number of records to see if it
is logarithmic, but doing so requires either longer data sets or a larger
collection of data sets, such as the combined readings of all stations in the
US.

The key observation is that the record highs and the record lows do not obey
the $1/t$ dependence. Hence, these temperature records must not samples from a
stationary process.

Wergen and Krug [25], [24], [9] and others, have taken this line of research
much further, to include correlations, a multiplicity of distributions,
consideration of spatial dependence. They have also found that removing a
trend in the data makes the theorem more likely to be consistent with the
statistics of temperature data. The values obtained for the trend, using this
analysis, are consistent with estimates of a rate of increase in the global
mean temperature of about 0.7oC in the land/ocean temperature over the last
century, roughly ten times faster than the average rate of ice-age-recovery
warming (see NOAA web site. Also see
https://globalclimate.ucr.edu/resources.html for educational material on this
topic).

In summary, the data suggests that Earth’s climate is a non-stationary
process. This is something climate scientists would find consistent with what
we know about climate. Hence, it is not likely that the temperature extremes
that we experience today are rare events, but rather, the result of a changing
climate. The findings of this line of inquiry, using much more technical
assumptions and allowing for correlations and for a multiplicity of
probability distributions, indicates that climate has been severely biased
upward during the Industrial Era. The use of this theorem to estimate the
return time of record high temperatures would seriously underestimate the
occurrence of historical highs, and overestimate historical lows.

## Incorporating Uncertainties in Greenhouse Gas (GHG) Projections

Global estimates of GHG emissions are readily available (see [3]) and their
estimates have tightly constrained uncertainties, since these are critical to
the economy of the energy sector. Private and public entities keep track of
production and resources fairly accurately. Uncertainties are associated,
however, with how to regulate and tax GHG. This uncertainty is financially
significant, but these would affect the design of commodity trading treaties,
just as uncertainties in many other commodities routinely affect how they are
traded. Uncertainties in the effect of GHG on climate play out locally: there
are well-recognized uncertainties related to transport and dispersion as well
as to critical dependencies on their interaction with their environment (land,
oceans, atmosphere). However, we are going to focus here on variability
spanning several decades to hundreds of years and the largest of spatial
scales. At these scales local nuances are not resolved and a balance model
yields the temporal evolution of a global temperature $T$.

By incorporating uncertainties into the temperature and radiative forcings we
can explore to what extent uncertainties in GHG affect conclusions of future
projections of the temperature. The uncertainties are derived from a
statistical analysis of the historical data. We can compare natural and
anthropogenic GHG forcings, taking into account uncertainties, in order to
determine whether the sensitivity of the outcomes depends on the relative
uncertainties in these two GHG components. We can also infer whether natural
or anthropogenic forcings are dominant, prior to the Industrial Era, during
this Era, and in the future.

Black body radiation tells us that the earth’s radiation is proportional to
$T^{4}$. The surface energy balance, in terms of the surface temperature $T$,
is $CdT/dt=Q+\kappa\sigma T_{Atm}^{4}-\sigma T^{4}$, where $T_{Atm}$ is the
atmospheric temperature, $t$ is time, $C$ is the effective heat capacity, and
$\sigma$ is Stefan-Boltzmann constant. $Q$ is the effective incoming
radiation. If $C_{a}dT_{Atm}/dt$ is small, where $C_{a}$ is the atmospheric
effective heat capacity, then $\kappa\sigma T^{4}+2\kappa\sigma
T^{4}_{Atm}\approx 0$, then $CdT/dt=Q-(1-\frac{\kappa}{2})\sigma T^{4}$. The
range of temperatures of the process is not large, hence,
$(1-\frac{\kappa}{2})\sigma T^{4}=A+BT$, is a linearization and the constants
in the nonlinear formula are subsumed by $A$ and $B$. The energy balance is
spectrally dependent: the high frequency component has a portion that reflects
back to space by clouds and snow/ice and one that mostly dissipates. The low
frequency component, on the other hand, is affected by reflectivity and the
presence of a complex layer of gas, dust, and droplets which is capable of
trapping the surface outgoing radiation. $Q$ will be assumed to be a linear
combination of the effective solar radiation and the radiative forcing due to
GHG. Hence $Q=\frac{1}{4}(1-\alpha)S+F_{GHG}$, where the albedo $\alpha\approx
0.3$, and the global average solar radiation $S/4\approx 1370/4$ Wm-2,
presently. (See [14], [17], and [12]). The Energy Balance Model (EBM) we adopt
is thus

$CdT=\frac{S}{4}(1-\alpha)dt+F_{GHG}dt-(A+BT)dt+\nu(t),$ (1)

where $T$ is the temperature of Earth’s surface (approximated as the surface
of a 70-meter-depth, mixed-layer ocean covering 70 percent of Earth’s surface
area). $C=2.08\times 10^{8}$ J K-1m-2 is the effective heat capacity that
accounts for the thermal inertia of the mixed-layer ocean, but does not allow
for heat exchange with the deep ocean as in more elaborate ’upwelling-
diffusion models’ ([26]). The last term in the equation is a stochastic
forcing term, added to represent inherent uncertainties and unresolved
processes (Hasselmann, [8], spearheaded the conceptualization of low
dimensional models for climate with an unresolved physics that had its own,
possibly stochastic, dynamic). The model (1) appears in [10] (and a link is
provided to matlab code which can be used to reproduce the temperature
outcomes in that article). The choice $A=221.3$ Wm-2 and $B=1.25$ WK-1m-2
yields a realistic preindustrial global mean temperature $T=14.8^{o}$C and an
equilibrium climate sensitivity (ECS) of 3.0 oK, consistent with midrange
estimates by the International Panel on Climate Change ([22]), as the range is
likely somewhere between $1.5-4.5$. The equilibrium climate sensitivity is
defined as the temperature change due to the effect of doubling in the
concentration of CO2 in Earth’s atmosphere. A $1^{o}$K change requires a
doubling of CO2, or equivalently, to an increased forcing of 3.7 W m-2.

The model was driven with estimated annual natural and anthropogenic forcing
over the years A.D. 850 to 2012. Greenhouse radiative forcing was calculated
using the approximation $F_{GHG}=5.35\log(\mbox{CO}_{2e}/280)$, where $280$
parts per million (ppm) is the pre-industrial CO2 level and CO2e is the
’equivalent’ anthropogenic CO2 (see [15]). The CO2 data from Ammann et al [1]
is used, scaled to give CO2e values 20% larger than CO2 alone (for example, in
2009 CO2 was 380 ppm whereas CO2e was estimated at 455 ppm). Northern
Hemisphere anthropogenic tropospheric aerosol forcing was taken instead from
Crowley [6], with an increase in amplitude of 5% to accommodate for a slightly
larger indirect effect than in [1]. A linear extrapolation of the original
series (which ends in 1999) is used to extend though 2012222One needs to
distinguish between emissions and CO2 concentrations. The latter are directly
measurable in the atmosphere and the increase is monotonic. This forms the
basis for the extrapolation model.. In the simulations we have assumed that
tropospheric aerosols decrease exponentially from their current values with a
time constant of 60 years. This gives a net anthropogenic forcing change from
2000 to 2100 of 3.5 Wm-2, roughly equivalent to the International Panel on
Climate Change’s 5th assessment report ’RCP6’ scenario, a future emissions
scenario that assumes only modest efforts at mitigation. Estimated past
changes in solar irradiance were prescribed as a change in the solar constant
$S$ whereas forcing by volcanic aerosols was prescribed as a change in the
surface albedo $\alpha$. Solar and volcanic forcing were taken from the
General Circulation Model (GCM) simulations described in McGuffie and
Henderson-Sellers [14]. These were modified as follows: the solar forcing was
rescaled under the assumption of a 0.1 percent change from Maunder Minimum to
present, more consistent with recent estimates ([1]); the volcanic forcing
applied was the mean of the latitudinally varying volcanic forcing in Ammann
et al [1]. With no added uncertainties, solar output and no climatically-
significant volcanic eruptions were assumed, for the years 2012-2100. CO2
radiative forcing has been extrapolated linearly, till the year 2100, based on
the trend over the past decade (which is roughly equivalent, from a radiative
forcing standpoint, to a forward projection of the exponential historical
trajectory of CO2 emissions). A net tropospheric anthropogenic aerosol forcing
rate decrease of 0.7 Wm-2 per year, starting with the year 2000 has been taken
into account in the model.

.

As summarized in the IPCC report the net rise in temperature during the
Industrial Era is due to the upward trend of the anthropogenic GHG forcing.
What’s more, there is no way to predict the temperature increases measured
during the Industrial Era, solely, by natural forcing. Figure 2 depicts EBM
temperature predictions as a function of various values of the equivalent
climate sensitivity (ECS) with historically-based additive uncertainties
added. Variations on ECS convey the sensitivity of the predictions to
epistemic uncertainties in the models. The instrumental record appears in
black in the figure. The temperature predictions shown represent single
realizations of the energy balance model. Details of the temperature noise
model appear in the Appendix.

Figure 2:  Temperature prediction, as a function of time, for various values
of equivalent climate sensitivity (ECS). Shown, a single realization for each
ECS. The stochastic model for the temperature was informed by instrumental
data. From top to bottom, $ECS=4.5,3,2.5,2,1.5$. The instrumental record
appears in black.

The outcome suggests that taking into account historically-informed
temperature variability in the predictions does not alter the conclusion that
the upward trend in the anthropogenic forcing remains dominant factor in
explaining temperature rises during the Industrial Era. We examine next
whether taking into account stochastic variability in the natural as well as
anthropogenic forcing changes this conclusion. Figure 3 portrays temperature
predictions from EBM model runs that incorporate uncertainties in the natural
forcing. In Figure 3a we include, exclusively, volcanic forcing uncertainties.
In Figure 3b we allow for solar forcing uncertainties. The various curves
result from using different ECS values. The description of the uncertainty
models for volcanic, solar, and CO2 source variability is found in the
Appendix. Plotting a single realization, it is hoped, conveys qualitatively
the relative impact of the stochasticity on the outcomes (qualitative
characteristics of the sample moments can be easily inferred from the specific
noise models used). As is expected, the temperature reacts abruptly to highly
localized volcanic emissions and

(a) (b)

Figure 3: Temperature predictions, as a function of time for various
equivalent ECS values. A single realization is depicted, and the same
stochastic model outcome is used in each of the ECS cases. In (a) predictions
are made by accounting, exclusively, for stochastic volcanic forcing
variability. In (b) for stochastic solar forcing variability exclusively. From
top to bottom, $ECS=4.5,3,2.5,2,1.5$. The instrumental record is depicted in
black.

the sensitivity to solar variability is small, comparatively. Hence, a
significantly larger portion of the variance in an ensemble of these
predictions is attributed to the volcanic emissions, rather than the solar
uncertainties.

Figure 4 shows the stochastic long wave and short wave forcing contributions
over time. The composite indicates a dominant role for the long wave component
during the last 50 years: the trend is larger than its inherent variability.

Figure 4: Stochastic long wave, short wave forcing and composite total
forcing, with uncertainties due to CO2 emissions, volcanic activity, and solar
forcing. A single stochastic realization is depicted.

In Figure 5 we depict a single realization of the temperature predictions,
driven by the forcing described in Figure 4, i.e., the temperature predictions
taking into account natural and anthropogenic forcing and their variability.
The upward trend in the long-wave emissivity still dominates over any
uncertainties due to natural and anthropogenic forcings during the Industrial
Era.

(a) (b)

Figure 5: (a) Highlight of the composite forcing appearing in Figure 4
corresponding to the period 1850-2100; (b) temperature predictions as a
function of ECS, taking into account uncertainties due to CO2 emissions,
volcanic activity, solar forcing. Stochastic variability due to temperature
uncertainties has been included. The stochastic model for temperature
fluctuations is informed by historical temperature variability data. From top
to bottom, $ECS=4.5,3,2.5,2,1.5$

In the end the steady-increasing CO2 forcing is overwhelming the temperature
predictions in the Industrial Era and into the future, even when taking into
account the variability due to volcanic and solar forcing.

## Summary

Without making use of anything more than data we have shown that temperatures
around the Northern Hemisphere do not have a time-stationary distribution.
More careful analysis of the data has shown that this is a general statement
regarding temperature most anywhere on Earth. Fits of the data, for Industrial
Era temperatures, show an upward trend. Ocean temperature surveys also
indicate that the temperature of the ocean in the upper 2000m has increased.
The signal-to-noise ratio of this Argo data, collected for the last 20 years,
is much higher than near-ground atmospheric temperatures [4]. In fact, data
and models indicate that the global mean temperature of the Earth is
increasing, since the end of the 19th century [18]. With a changing climate,
we have observed a changing weather. The rate at which climate is changing is
alarming since it is comparable to, or shorter than, the typical relaxation
rates of the system (land, ocean, and atmosphere).

An energy balance model, consistent with instrumental data, was used here to
explore how the inclusion of inherent uncertainties affect the relative impact
of natural and anthropogenic forcing on Earth’s temperature [10]. We estimated
uncertainties from the data and constructed historically-informed models for
the variability of each of the forcings. Both of these forcings need to be
invoked to get the model to agree with data. Driven by estimated natural and
anthropogenic radiative forcing, our calculations indicate that the warming is
a result of anthropogenic increases in greenhouse gas concentrations and that
the inclusion of aleatoric uncertainties do not change this outcome in the
ensemble sense. Moreover, since the effect of forcing variability is small
compared to the upward trend of the anthropogenic forcing, the inherent
variability does not offer much of a reason to expect that the temperature
will keep climbing unless there is a rate slowdown in the anthropogenic
forcing, i.e., in the production of GHG. Using far more sophisticated models
and state-of-the-art knowledge in climate, scientists reach the same
conclusions [22] and further, have not been able to find a non-anthropogenic
explanation to the observed increase in warm extremes in global temperatures
during the Industrial Era (e.g., [11]).

The key natural forcings are associated with volcanic emissions and changes in
insolation. While statistical projections of changes in natural volcanic and
solar radiative forcing of climate are by necessity speculative, we find no
evidence that such natural radiative forcing changes could substantially alter
the projected warming from increasing greenhouse gas concentrations. The
impact of their variability would contribute to known unknowns in the
temperature uncertainty. The long-time features of the model and the
historical data agree well and thus do not require postulating or requiring
epistemic variability (unknown unknowns).

Shah’s appraisal of the outcomes in the Fourth National Assessment report
offered us an invitation to demonstrate how simple, well established,
quantitative methods are used to address apparent challenges posed by
uncertainties in climate assessments. Given the evidence that key climate
change attributes, such as ice sheet collapse and sea level rise, are
occurring ahead of schedule [13] uncertainty has in many respects broken
against us, rather than in our favor. Scientific uncertainty is not a reason
for inaction. If anything, it is a reason for more concerted efforts to limit
carbon emissions

## Acknowledgements

We want to thank Barbara Levi who provided invaluable editorial assistance
with this article.

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## Appendix, Stochastic Parameterizations

We describe the parametrization of the natural and anthropogenic forcing, used
in the future projections produced with the EBM model, (1). The process for
the stochastic parametrization follows the strategy used in [20].

Volcanic Emissions: Using historical data from [1], we propose the following
stochastic model:

$dy(t)=-b\,y(t)dt+\exp{({\cal J}_{t})}dP_{t},$

where $b=260/365$, ${\cal J}_{t}$ is an exponential process with uniform
random argument $1.2\,{\cal U}[0,1]$, and $dP_{t}$ is a Poisson incremental
process with $\lambda=0.07$. Figure 6 shows a comparison of the data (lower),
and the negative of the stochastic simulation (upper).

Figure 6: Lower: volcanic forcing data; upper, stochastic parametrization.

Solar Forcing: The solar data for the years 850-2012 AD from [1] is used. In
what follows, the mean 1365.8 W/m2 is removed. Figure 7a displays the modulus
of the spectral decomposition of the (periodized) observations. Figure 7b
features the observations and the parametrization, along with the intrinsic
time decomposition of the signal [19].

(a)
(b)  (c)

Figure 7: Solar forcing (W/m2). The historical mean 1366 W/m2 is subtracted.
(a) Spectrum of the observations; (b) comparison of data and the
parametrization of the solar forcing. Data (*), parametrization (o). The
baseline (blue) and the rotation (red) from the intrinsic time decomposition;
(c) observations (o) and extrapolated (*) solar forcing.

The intrinsic time decompostion of the signal $s(t)=b(t)+r(t)$, is not lossy.
The baseline $b(t)$ is designed to lace through the data in such a way that
local maximas (minimas) of the time series $s(t)$ are below (above) the
baseline at the locations of the extremas. At the extremal locations the
baseline is situated at the three-point average of the extreme and its two
neighbors. The rest of the baseline is constructed by linear interpolation.
The rotation signal is defined by $r(t)=s(t)-b(t)$. The parametrization
$\sigma(t)=b(t)+n(t)$, where $n(t)$ is an ARMA(4,0) process obtained from the
rotation $r(t)$. In the ARMA, the constant was -0.0010598, $p_{1}=1.7193$,
$p_{2}=-1.192$, $p_{3}=1.0518$, $p_{4}=-0.58531$. The standard error on these
was less than one percent. The extrapolation of $\sigma(t)$ beyond the
observations, i.e., beyond 2012, is obtained by a reset of time in
$b(t-t_{r})$, where $t_{r}$ is drawn at random from the years 928 and 1924.
The value of the solar observations at 2012 is used to adjust the
extrapolation. The baseline assumed free boundary conditions at the ends of
the observation record. The time scales in the ARMA process was large enough
that, for short extrapolations of the signal, in the interval 2012-2100, it
was impossible to discern the persistence of the baseline signal. Figure 7c
depicts the signal plus its extrapolation beyond year 2012.

CO2 Forcing Rate: We use a deterministic fit $\kappa_{i}(t)$, obtained by
least squares, from the CO2 forcing signal. There are two fitted regions,
namely,

$\kappa_{1}(t)=\sum_{i=0}^{4}a_{i}\tilde{y}^{i},$

where $\tilde{y}$ = year-1770, with parameters $a_{4}=-1.59\times 10^{-7}$,
$a_{3}=5.93\times 10^{-5}$, $a_{2}=-0.006$, $a_{1}=0.265$ and $a_{0}=278$, for
the years $1770-1955$, and

$\kappa_{2}(t)=\sum_{i=0}^{2}a_{i}\tilde{y}^{i},$

where $\tilde{y}=$ year-1956, for the span $1956-2012$, with parameters
$a_{2}=0.012$, $a_{1}=0.7943$, and $a_{0}=394.7128$. We combine the
variability $T(t)-\kappa_{1}(t)$ of $1770-1955$, and $T(t)-\kappa_{1}(t)$ of
$1956-2012$, and propose an AR(1) model (constant = -0.077584 and
$p_{1}=0.80513$), with variance $0.17$ yr2. We then use $\kappa_{2}(t)$ to
extrapolate to the year 2100 AD, adding the AR(1) process for variability. See
Figure 8.

Figure 8: Superposition of the CO2 forcing rate data and the stochastically-
fitted version; in red, stochastic parametrization for the years $2012-2100$.

Parametrizing Future Temperature Uncertainty:

The temperature uncertainty $\nu(t)$ is parametrized as follows. We use
compilations of the Northern Hemisphere temperatures, for the years 1850-2012
AD (see [10]). An ARIMA analysis of the raw data was performed, double
differencing the data. The outcomes were satisfactory, but not as robust as
using a fitted removal of the trend and a noise model for the residual. The
parametrization consists of a trend $\tau$ plus a stochastic parametrization
of $T-\tau$, where $T$ is the normalized temperature record. The fit of the
trend is

$\tau=\sum_{i=0}^{3}a_{i}\tilde{y}^{i}$

where $\tilde{y}$=year-1850, and $a_{3}=2.644\times 10^{-7}$,
$a_{2}=-7.98\times 10^{-6}$, $a_{1}=-0.000167$, $a_{0}=-0.274$. The residual
$T-\tau$ is represented well (according to the Bayesian Index Criteria, BIC),
by an AR(1), with constant $=-3.0333\times 10^{-5}$, $p_{1}=0.53426$, with
errors in the order of 6 percent.

(a)  (b)

Figure 9: (a) Fit of the trend $\tau=\sum_{i=0}^{3}a_{i}\tilde{y}^{i}$, where
$\tilde{y}=$year$-1850$, and data to the temperature distribution; (b)
stochastic parametrization and data.