# Convergence properties of symmetrization processes

Jacopo Ulivelli ***Affiliation: La Sapienza, University of Rome, Italy,
Piazzale Aldo Moro 5, 00185. Email<EMAIL_ADDRESS>
###### Abstract

Steiner symmetrization is well known for its rounding and general convergence
properties. We identify a whole family of symmetrizations sharing analogue
behaviors: In fact we prove that all these symmetrizations share the same
converging symmetrization processes, together with some pathological
phenomena. †††Mathematics Subject Classification. Primary: 52A20,52A38.
Secondary: 52A30,51F15,52A39. Key words and phrases. convex body, Steiner
symmetrization, Schwarz symmetrization, Minkowski symmetrization, fiber
symmetrization, Minkowski addition, universal sequences.

## 1 Introduction

Symmetrizations play a very important role in geometry and its application,
allowing to prove many results with relatively easy and direct proofs. In
particular geometric and analytic inequalities, like the Isoperimetric,
Blaschke-Santalò, Faber-Krahn inequalities and many others have been proved
using for example Steiner symmetrization. For some self contained
introductions on the subject see [9] Chapter 1 and 2, [10] Chapter 9, [17]
Chapter 10, [12] Chapter 3 and the references therein. All these results rely
mainly on the fact that through Steiner symmetrization, for every convex
compact set it is always possible to find a sequence of symmetrizations
converging to a ball while preserving the volume.

We call symmetrization process a sequence of symmetrizations applied to a
subset of $\mathbb{R}^{n}$. These processes are the main focus of this work.
In 1986 Mani-Levitska [14] showed that, for Steiner symmetrization, a randomly
chosen symmetrization process for a convex compact body converges almost
surely to a ball. This result was later extended by Van Schaftingen [16] to
general compact sets, then by Volcic [19] for measurable sets. Coupier and
Davydov [7] later proved, thanks to the inclusion between Steiner and
Minkowski symmetrization, that an analogue probabilistic property holds for
Minkowski symmetrization. Other interesting results concerning convergence in
probability can be found in the works of Bianchi, Burchard, Gronchi and Volcic
[1] and Burchard and Fortier [6].

Parallelly, the study of deterministic convergence received a boost in 2012
with the work of Klain [13], which inspired a series of papers on the subject.
In particular Bianchi, Gardner and Gronchi in [2] and [3] introduced a general
framework for the study of symmetrizations, focusing, among the others, on the
relations between different symmetrizations and their properties. We mainly
use their formalism, which now we introduce. Let $\mathcal{E}$ be a family of
sets (they might be convex compact sets, epigraphs of a certain class of
functions…) and fix a subspace $H$, then a $H$-symmetrization is a map

$\diamondsuit_{H}:\mathcal{E}\to\mathcal{E}_{H},$

where $\mathcal{E}_{H}$ is the subfamily of $\mathcal{E}$ of $H$-symmetric
sets. Here we focus on the family of compact sets $\mathcal{C}^{n}$ and convex
compact sets $\mathcal{K}^{n}$. Symmetrizations may enjoy many different
properties, some of which can completely characterise them, as it is shown for
example in [2], Section 9.

The properties we are interested in are monotonicity, invariance under
reflections and invariance under orthogonal translations (see next Section for
the specific definitions). We call $\mathcal{F}$ the family of all the
symmetrizations satisfying such properties for every proper subspace of
$\mathbb{R}^{n}$. The family $\mathcal{F}$ includes Minkowski and fiber
symmetrizations (the latter corresponds for convex compact bodies to the
Steiner symmetrization when considered with respect to hyperplanes) and
possesses a very strong structure, as proved in Corollary 7.3 in [2]. Such
structure plays an essential role in the proof of our main results. Indeed it
will allow us to characterise some convergence phenomena for $\mathcal{F}$
which are shared by the whole family. A convergence property that has been
studied in literature is the following.

######  Definition 1.1.

If $\diamondsuit$ is a symmetrization on $\mathcal{K}^{n}$, a sequence of
subspaces $(H_{m})$ of $\mathbb{R}^{n}$ is said to be weakly-universal for
$\diamondsuit$ if, for every $k\in\mathbb{N}$ we have that the sequence of
sets

$K_{m,k}=\diamondsuit_{H_{m}}\dots\diamondsuit_{H_{k}}K$

converges for every $K\in\mathcal{K}^{n}$ with non empty interior to a ball of
radius $r(K,k)$, thus we allow such quantity to change with respect to $k$. If
$r(K,k)$ is independent from $k$, then $(H_{m})$ is said to be universal for
$\diamondsuit$.

The definition of universal sequence was introduced in [7] (Theorem 3.1), were
the following result was obtained for the family $\mathcal{K}^{n}_{n}$ of
convex bodies, i.e. convex compact subsets of $\mathbb{R}^{n}$ with non empty
interior.

######  Theorem 1.2 (Coupier and Davydov).

A sequence of hyperplanes $(H_{m})$ in $\mathbb{R}^{n}$ is universal for
Steiner symmetrization in $\mathcal{K}^{n}_{n}$ if and only if it is universal
for Minkowski symmetrization in $\mathcal{K}^{n}_{n}$.

Later this Theorem was extended by Bianchi, Gardner and Gronchi together with
the introduction of weakly universal sequences. In [2], section 8 and [3]
section 6 and 7 many results were obtained in this direction, in particular we
observe that Theorems 7.3 and 7.4 from [3] together with Theorem 1.2
generalize the latter result to the family $\mathcal{C}^{n}$.

In all these results the limit of the sequence is a ball. Here instead we
study a wider class of sequences, without prescribing specific limit sets.

######  Definition 1.3.

If $\diamondsuit$ is a symmetrization on $\mathcal{E}$, a sequence of
subspaces $(H_{m})$ is said to be $\diamondsuit$-stable (or stable for the
symmetrization $\diamondsuit$) if, for every $k\in\mathbb{N}$ the sequence
defined for $m\geq k$

$K_{m}=\diamondsuit_{H_{m}}\dots\diamondsuit_{H_{k+1}}\diamondsuit_{H_{k}}K$

converges for every $K\in\mathcal{E}$.

Notice that the limit might depend on $K\in\mathcal{K}^{n}$ and
$k\in\mathbb{N}$ both, as we show in the Examples 3.2 and 3.3.

The generalization of Theorem 1.2 we prove is the following.

######  Theorem 1.4.

Let $\diamondsuit\in\mathcal{F}$ be a symmetrization on $\mathcal{K}^{n}$.
Then, if $(H_{m})$ is a $\diamondsuit$-stable sequence of subspaces of
$\mathbb{R}^{n}$, it is $\spadesuit$-stable for every symmetrization
$\spadesuit\in\mathcal{F}$.

In particular this holds for Steiner and Minkowski symmetrizations when
$(H_{m})$ are hyperplanes.

This turns out to be a simple consequence of a broader result we prove in
Theorem 3.4, which involves the so called convergence in shape, studied for
the first time in [1]. This kind of convergence, properly defined in the next
section, involves a sequence of rotations $(\mathbb{A}_{m})$, which corrects
at every step the underlying symmetrization process. Indeed such tool can be
used to give convergence to symmetrizations processes which would not
naturally have a limit, as the ones we study in Section 4. Using the same
notation of Theorem 1.4, we prove in Theorem 3.4 that, given two
symmetrizations $\diamondsuit,\spadesuit\in\mathcal{F}$, we have that
$\diamondsuit$ is stable in shape if and only if $\spadesuit$ is stable in
shape.

This last result has many interesting consequences. One of them is a partial
answer to the question: Does a converging sequence of hyperplanes induce a
converging symmetrization process? We find a positive answer when one
additional assumption is imposed.

######  Theorem 1.5.

Let $(H_{m})$ be a sequence of hyperplanes and consider the corresponding
normals $(u_{m})\subset\mathbb{S}^{n-1}$. Consider moreover
$\diamondsuit\in\mathcal{F}$ . If the angles $(\alpha_{m})\subset[0,\pi/2]$
given by the relation $\absolutevalue{u_{m}\cdot u_{m-1}}=\cos\alpha_{m}$ are
such that

$\sum_{m\in\mathbb{N}}\absolutevalue{\alpha_{m}}<+\infty,$

then the sequence $(H_{m})$ converges and it is $\diamondsuit$-stable on
$\mathcal{K}^{n}$.

We were not able to prove that such assumption is necessary, but the question
is the subject of ongoing research.
The structure of this work is the following. In Section 2 we introduce the
basic notation and definitions, recalling some tools and instrumental results.
In Section 3 we prove Theorem 3.4 and investigate some consequences. Finally
in Section 4, after presenting a counterexample from [1], we prove new ones
and show that the same procedures work for all the symmetrizations of the
family $\mathcal{F}$.

## 2 Preliminaries

Our ambient space is the family of compact subsets of $\mathbb{R}^{n}$, where
we consider the following operation: Given two subsets $A,B$ of
$\mathbb{R}^{n}$, the Minkowski addition of $A$ and $B$ is the set

$A+B:=\\{x+y|x\in A,y\in B\\}.$

Such space is a complete metric space with respect to the topology induced by
the Hausdorff distance, which for two compact sets $K,L$ is given by

$d_{\mathcal{H}}(K,L):=\max\\{\inf\\{\varepsilon>0,K\subset L+\varepsilon
B^{n}\\},\inf\\{\varepsilon>0,L\subset K+\varepsilon B^{n}\\}\\},$

where $B^{n}$ is the Euclidean $n$-dimensional unitary ball centered in the
origin. See [17], [12], [11] for the classical theory of convex bodies.

We denote as $\mathcal{C}^{n}$ and $\mathcal{K}^{n}$ the families of compact
sets and convex compact sets of $\mathbb{R}^{n}$ respectively. When using the
subscripts $\mathcal{C}^{n}_{n},\mathcal{K}^{n}_{n}$, we are restricting
ourselves to the respective families of bodies, i.e. sets with non-empty
interior. Throughout the paper $B(x,r)$ denotes the Euclidean ball centred in
$x$ of radius $r>0$, and we write the Euclidean scalar product between two
vectors $x,y\in\mathbb{R}^{n}$ as $x\cdot y$. $\mathbb{S}^{n-1}$ is the unit
sphere in $\mathbb{R}^{n}$.

When dealing with symmetrizations a crucial role is played by some specific
set functions on the family of sets we are considering. In particular, for
$\mathcal{C}^{n}$ we are interested in the volume, given by the
$n$-dimensional Lebesgue measure of a set which we write as
$\lambda_{n}(\cdot)$. For convex sets we can also consider the mean width,
which for $K\in\mathcal{K}^{n}$ is

$\mathcal{W}(K):=\frac{1}{\omega_{n}}\int_{\mathbb{S}^{n-1}}[h_{K}(\nu)+h_{K}(-\nu)]d\mathcal{H}^{n-1}(\nu),$

where $\omega_{n}$ is the $(n-1)$-dimensional measure of $\mathbb{S}^{n-1}$
and $h_{K}$ is the support function of $K$, given by

$h_{K}(x):=\sup\\{x\cdot y|y\in K\\},\quad x\in\mathbb{S}^{n-1}.$

An important property of the Minkowski addition of compact convex sets is that
it is equivalent to the sum of support functions in the following sense:

$h_{K}(\cdot)+h_{L}(\cdot)=h_{K+L}(\cdot)$ (2.1)

for every $K,L\in\mathcal{K}^{n}$. In this notation, the translation of a set
$A$ by a vector $x\in\mathbb{R}^{n}$ may be written as $A+x$.
The symmetric difference of two measurable subsets $A,B$ of $\mathbb{R}^{n}$
is

$\lambda_{n}(A\Delta B):=\lambda_{n}(A\setminus B)+\lambda_{n}(B\setminus A).$

In $\mathcal{K}^{n}_{n}$ this is the Nikodym distance, which is equivalent to
the Hausdorff distance.

We denote as $\mathcal{G}(n,i),1\leq i\leq n-1$ the sets of subspaces of
$\mathbb{R}^{n}$ of dimension $1\leq i\leq n-1$. We use the term subspace
referring to linear subspaces. For every subset $A$ of $\mathbb{R}^{n}$ and
subspace $H$ we writethe projection of $A$ onto $H$ as $P_{H}A$. The
reflection with respect to a subspace $H$, given by the map

$x\mapsto x-2P_{H^{\perp}}\\{x\\},$ (2.2)

is denoted by $R_{H}$, where $H^{\perp}$ is the subspace orthogonal to $H$.
When a set $A$ of $\mathbb{R}^{n}$ is such that $A=R_{H}A$, it is said to be
symmetric with respect to $H$, or $H$-symmetric for short.

A fundamental result in the theory of convex bodies is the Brunn-Minkowski
inequality. It states that for every compact sets $K,L$ the inequality

$\lambda_{n}(K+L)^{1/n}\geq\lambda_{n}(K)^{1/n}+\lambda_{n}(L)^{1/n}$ (2.3)

holds, where equality is achieved if and only if $K$ and $L$ are homothetic
convex bodies or lower-dimensional convex sets liyng in two parallel affine
subspaces. For a complete survey regarding this inequality and its extensions
see Gardner [8].

We now go back to the concept of $H$-symmetrization introduced in [2], [3].
Given a family of sets $\mathcal{E}$ and a subspace $H$, an $H$-symmetrization
is a map

$\diamondsuit_{H}:\mathcal{E}\to\mathcal{E}_{H}$

with $\mathcal{E}_{H}=\\{C\in\mathcal{E}|R_{H}E=E\\}$, where $R_{H}$ is the
map given by (2.2).
Let $\diamondsuit$ be a $H$-symmetrization in $\mathcal{E}$. Of particular
interest are the following properties:

  1. 1.

(Monotonicity): For every $K,L\in\mathcal{E}$ if $K\subseteq L$, then
$\diamondsuit_{H}K\subseteq\diamondsuit_{H}L$,

  2. 2.

(Idempotence): For every $K\in\mathcal{E}$ holds
$\diamondsuit_{H}K=\diamondsuit_{H}\diamondsuit_{H}K$,

  3. 3.

($H^{\perp}$-translation invariance for $H$-symmetric sets): If
$K\in\mathcal{E}$ and $R_{H}K=K$, then for every $x\in H^{\perp}$ holds
$\diamondsuit_{H}(K+x)=K$,

  4. 4.

(Invariance for $H$-symmetric sets): If $K\in\mathcal{E}$ and $R_{H}K=K$,
$\Rightarrow\diamondsuit_{H}K=K$.

  5. 5.

($F$-invariance): There exist a function $F:\mathcal{E}\to\mathbb{R}$ such
that $F(K)=F(\diamondsuit_{H}K)$ for every $K\in\mathcal{E}$.

When we refer to $\diamondsuit$ as a symmetrization it is understood that
$\diamondsuit_{H}$ is a $H$-symmetrization for every subspace $H$ of
$\mathbb{R}^{n}$. Of particular interest is the following family of
symmetrizations:

$\mathcal{F}:=\\{\diamondsuit\text{ symmetrization}|\text{ properties 1,3,4
hold }\\}.$

The results obtained in this paper concern mainly Schwarz, Minkowski e fiber
Symmetrizations, which we proceed now to present.

### Schwarz Symmetrization

Let $C\in\mathcal{C}^{n}$, for a fixed $H\in\mathcal{G}(n,i),1\leq i\leq n-1$.
The Schwarz symmetrization of $C$ is the set

$S_{H}C:=\bigcup_{x\in H}B(x,r_{x}),$

where $r_{x}$ is such that
$\lambda_{n-i}(K\cap(H^{\perp}+x))=\lambda_{n-i}(B(x,r_{x}))$ if
$\lambda_{n-i}(K\cap(H^{\perp}+x))>0$. If the measure of the section a t $x\in
H$ is zero but the section is non empty, we replace it with $x$, otherwise we
replace such section with the empty set. From Fubini’s Theorem it follows that
this symmetrization preserves the volume, thus satisfying property (5) for
$F(\cdot)=\lambda_{n}(\cdot)$. When $i=n-1$ this is better know as Steiner
Symmetrization, and in general it decreases intrinsic volumes (see [11], Satz
XI, p. 260, or Theorem 10.4.1 in [17]). Both in $\mathcal{C}^{n}$ and
$\mathcal{K}^{n}$ Schwarz symmetrization satisfies properties (1,2,5), while
(3,4) hold only for convex sets in the case $i=n-1$.

### Minkowski Symmetrization

Let $C\in\mathcal{C}^{n}$, $H\in\mathcal{G}(n,i),1\leq i\leq n$. The Minkowski
symmetrization of $C$ is the set

$M_{H}K:=\frac{C+R_{H}C}{2}.$

Clearly from the definition of $\mathcal{W}(K)$ and (2.1) it preserves the
mean width when $C$ is convex, thus property (5) holds for
$F(\cdot)=\mathcal{W}(\cdot)$. From the Brunn-Minkowski inequality (2.3) it
follows that

$\lambda_{n}(M_{H}K)\geq\lambda_{n}(K).$

It may be useful to consider the central Minkowski symmetrization, i.e.

$M_{o}K=\frac{K-K}{2},$

which is centrally symmetric. If $K$ lies in an affine subspace $H+x,x\in
H^{\perp}$, then we write $M_{x}K$ for the central Minkowski symmetrization in
such affine subspace, i.e.

$M_{x}K=\frac{K+R_{H^{\perp}}K}{2}.$

In $\mathcal{K}^{n}$ Minkowski symmetrization satisfies all the listed
properties, but in $\mathcal{C}^{n}$ only (1) holds.

### Fiber Symmetrization

Let $C\in\mathcal{C}^{n}$, $H\in\mathcal{G}(n,i)1\leq i\leq n$. Then its fiber
symmetrization is the set

$F_{H}K=\bigcup_{x\in H}M_{x}(K\cap(H^{\perp}+x)).$

This symmetrization can be seen as an hybrid between Schwarz and Minkowski
symmetrization, and the underlying operation was introduced by McMullen [15].
As Minkowski symmetrization it increases the volume and in $\mathcal{K}^{n}$
it satisfies property (1,2,3,4), while (2,3,4) fail to hold in
$\mathcal{C}^{n}$. See [18] for some specific examples.

Fiber and Minkowski symmetrizations can be considered the extremals of the
family $\mathcal{F}$, in a sense that Theorem 2.1 will make clear. Moreover
other symmetrizations, different from fiber or Minkowski, can be found in
$\mathcal{F}$; see for example the remarks after Corollary 7.3 in [2], where
the following fact is proved.

######  Theorem 2.1 (Bianchi, Gardner and Gronchi).

Let $H\in\mathcal{G}(n,i),1\leq i\leq n-1$, $\mathcal{E}=\mathcal{K}^{n}$ or
$\mathcal{K}_{n}^{n}$. If $\diamondsuit\in\mathcal{F}$, then

$F_{H}K\subseteq\diamondsuit_{H}K\subseteq M_{H}K$ (2.4)

for every $K\in\mathcal{E}$.

In the last section of this paper we study some counterexamples to the
convergence of symmetrization processes. We will present one in Example 4.1,
which was independently proved in [5] and [6]. Such example shows that a dense
sequence of directions, if accurately chosen, leads to a non converging
symmetrization process. We show in particular that the same construction holds
for the whole family of symmetrizations considered in Theorem 2.1.

For this family of non converging sequences, when dealing with Steiner
symmetrization of compact sets, convergence is still possible in a weaker
sense, called convergence in shape. The next section will be devoted to the
study and the generalization of such convergence (see Definition 3.2). The
first result in this direction was achieved in [1], Theorem 2.2, with the
following result.

######  Theorem 2.2 (Bianchi, Burchard, Gronchi and Volcic).

Let $(u_{m})$ be a sequence in $\mathbb{S}^{n-1}$ such that $u_{m}\cdot
u_{m-1}=\cos\alpha_{m}$, where $(\alpha_{m})$ is a sequence that satisfies
$\sum_{m\in\mathbb{N}}\alpha_{m}^{2}<+\infty$. Define moreover $(H_{m})$ as
the corresponding sequence of orthogonal hyperplanes given by
$H_{m}=u_{m}^{\perp}$ for every $m\in\mathbb{N}$.

Then there exist a sequence of rotations $R_{m}$ such that for every non empty
compact set $K\subset\mathbb{R}^{n}$ the sequence

$K_{m}:=R_{m}S_{H_{m}}\dots S_{H_{1}}K$

converges in Hausdorff distance to a convex compact set L.

In Corollary 3.6 we obtain a generalization of such result in
$\mathcal{K}^{n}$ for all the symmetrizations in $\mathcal{F}$ with respect to
sequences of hyperplanes.

## 3 Shape-Stable Symmetrization Processes

We start noticing some monotonicity properties for volume and mean width with
respect to symmetrizations in $\mathcal{F}$.

######  Lemma 3.1.

Consider $H\in\mathcal{G}(n,i),1\leq i\leq n-1,$ and a symmetrization
$\diamondsuit_{H}:\mathcal{K}^{n}\to(\mathcal{K}^{n})_{H}$ such that
$\diamondsuit\in\mathcal{F}$. Then for every $K\in\mathcal{K}^{n}$ hold

$\lambda_{n}(\diamondsuit_{H}K)\geq\lambda_{n}(K),\quad\mathcal{W}(K)\geq\mathcal{W}(\diamondsuit_{H}K).$

###### Proof.

The first inequality is a consequence of the Brunn-Minkowski inequality (2.3).
Indeed by definition, if $K\in\mathcal{K}^{n}$ then every $H$-orthogonal
section of $F_{H}K$ is a Minkowski symmetrization of a $H$-orthogonal section
of $K$, thus $\lambda_{n}(F_{H}K)\geq\lambda_{n}(K)$.
Now, thanks to the inclusions chain (2.4) we have

$\lambda_{n}(\diamondsuit_{H}K)\geq\lambda_{n}(F_{H}K)\geq\lambda_{n}(K).$

For the second inequality, again thanks to (2.4) we have that
$\diamondsuit_{H}K\subseteq M_{H}K$, thus clearly
$h_{\diamondsuit_{H}K}(u)\leq h_{M_{H}K}(u)$ for every $u\in\mathbb{S}^{n-1}$
and consequently

$\mathcal{W}(\diamondsuit_{H}K)\leq\mathcal{W}(M_{H}K)=\mathcal{W}(K),$

completing the proof. ∎

We need now a weaker concept of convergence, which generalizes the phenomenon
studied in Theorem 2.2.

######  Definition 3.2 (Convergence in Shape).

Given a set $C\in\mathcal{E}$, a symmetrization $\diamondsuit$ on
$\mathcal{E}$ and a sequence of subspaces $(H_{m})$, the sequence of
symmetrals

$\diamondsuit_{H_{m}}\dots\diamondsuit_{H_{1}}C$

is said to converge in shape if there exist a sequence of rotations
$(\mathbb{A}_{m})$ such that

$\mathbb{A}_{m}\diamondsuit_{H_{m}}\dots\diamondsuit_{H_{1}}C$ (3.1)

converges.

######  Definition 3.3 (Shape-stable Sequences).

If in Definition 3.2 for every $k\in\mathbb{N},m\geq k$ the sequence

$\mathbb{A}_{m}\diamondsuit_{H_{m}}\dots\diamondsuit_{H_{k}}C$

converges, where $(\mathbb{A}_{m})$ is independent from $C$ and $k$, the
sequence of subspaces $(H_{m})$ is shape-stable in $\mathcal{E}$ for
$\diamondsuit$.

Examples of a shape-stable sequences are presented in Theorem 2.2. Notice that
if $\mathbb{A}_{m}$ is the identity for every $m\in\mathbb{N}$, then a shape-
stable sequence $(H_{m})$ is stable. To better understand these new concepts,
let us show some examples.

######  Example 3.1 (Klain’s Theorem).

In [13] Klain proved that, considered a finite family of hyperplanes
$\mathcal{D}=\\{G_{1},\dots,G_{k}\\}$, a sequence $(H_{m})$ such that
$H_{m}\in\mathcal{D}$ for every $m\in\mathbb{N}$ is stable for Steiner
symmetrization on $\mathcal{K}^{n}$. This result is extended in [3] to
Minkowski, Fiber, Schwarz and more generic symmetrizations, again in the
family $\mathcal{K}^{n}$. In [18] we proved that such result holds as well for
the Minkowski symmetrization on compact sets of $\mathbb{R}^{n}$.

######  Example 3.2 (Stable Sequence).

Consider in $\mathbb{R}^{2}$ the square $Q$ with vertices $(1,0)$, $(0,-1)$,
$(-1,0)$, $(0,1)$ as in Figure 1. Consider the sequence of lines $(H_{m})$
where $H_{m}=span\\{(1,0)\\}$ when $m$ is even, $H_{m}=span\\{(0,1)\\}$ when
$m$ is odd, for $m\geq 2$, while $H_{1}=span\\{(\sqrt{2}+1,1)\\}$. Then
$(H_{m})$ is stable in $\mathcal{K}^{2}$ for Minkowski symmetrization thanks
to Theorem 5.7 [3] (see example 3.1). Now, observe that $M_{H_{1}}Q$ is the
red octagon in the figure, and all the other symmetrizations leave this body
unchanged, so that the limit is be exactly $M_{H_{1}}Q$. If we start from
$m\geq 2$ instead the limit is always $Q$.

$H_{2},H_{4},H_{6},\dots$$H_{3},H_{5},H_{7},\dots$$H_{1}$ Figure 1: Different
limits may arise from stable sequences.

######  Example 3.3 (Shape-stable Sequence).

Consider in $\mathbb{R}^{2}$ an ellipse $E$ centered in the origin and a
sequence of lines as in Theorem 2.2. It is known that Steiner symmetrization
preserves ellipses and we can choose a direction $v$ such that the
symmetrization with respect to the line $H_{1}$ parallel to $v$ gives a ball.

If we consider a sequence of lines $(H_{m})$ starting from $H_{1}$ and then
continuing as the sequence of Theorem 2.2, we infer that $(H_{m})$ is shape-
stable in $\mathcal{K}^{2}$. Moreover, considering that $S_{H_{1}}E$ is a ball
centered in the origin the limit is of course $S_{H_{1}}E$. If we skip the
first symmetrizations, as it was proved in Example 2.1 from [1] (which we
recall here in Example 4.1) we can choose the remaining directions such that
the limit of the convergence in shape is not a ball.

We can now prove the following equivalence result. Notice that for this result
the dimension of the subspaces in the sequence is not relevant.

######  Theorem 3.4.

Let $\diamondsuit\in\mathcal{F}$. Given a sequence of subspaces $(H_{m})$
shape-stable for $\diamondsuit$ in $\mathcal{K}^{n}$ with isometries
$(\mathbb{A}_{m})$, $(H_{m})$ is shape-stable in $\mathcal{K}^{n}$ with
isometries $(\mathbb{A}_{m})$ for every symmetrization
$\spadesuit\in\mathcal{F}$.

In particular a sequence of subspaces $(H_{m})$ is shape-stable for
$\diamondsuit\in\mathcal{F}$ if and only if the same property holds for fiber
or Minkowski symmetrizations. If $(H_{m})$ are hyperplanes, the same
conclusion holds for Steiner symmetrization.

###### Proof of Theorem 3.4.

The outline of the proof is the following. We will proceed applying multiple
times (2.4), first proving that if $(H_{m})$ is shape stable for
$\diamondsuit$ in $\mathcal{K}^{n}$ then it is shape-stable for $M$ in
$\mathcal{K}^{n}$. After that, we show that if the same sequence is shape-
stable for $M$ in $\mathcal{K}^{n}$ then the same holds for $\spadesuit$.

Let $(H_{m})$ be a shape-stable sequence of subspaces for $\diamondsuit$ in
$\mathcal{K}^{n}$ as in the hypothesis, we want to prove that for every
$K\in\mathcal{K}^{n}$ the sequence

$K_{m}=\mathbb{A}_{m}M_{H_{m}}\dots M_{H_{1}}K$ (3.2)

converges. Suppose by contradiction that there exists $K\in\mathcal{K}^{n}$
such that for two subsequences $K_{m_{j}},K_{m_{l}}$ obtained by (3.2) one has
$K_{m_{j}}\to L_{1},K_{m_{l}}\to L_{2}$ where $L_{1}\neq L_{2}$,
$L_{1},L_{2}\in\mathcal{K}^{n}$.

Consider the sequence of bodies originated from the same process starting from
$K_{r}:=K+B(0,r)$ instead of $K$, $r>0$ fixed. This is done in order to cover
both the full and lower dimensional cases at the same time.
Notice that if $H$ is a subspace and $\mathbb{A}$ is a rotation, for every
$K\in\mathcal{K}^{n}$

$M_{H}(K+B(0,r))=M_{H}(K)+B(0,r),\quad\mathbb{A}(K+B(0,r))=\mathbb{A}(K)+B(0,r)$

thus

$(K_{r})_{m}:=\mathbb{A}_{m}M_{H_{m}}\dots
M_{H_{1}}(K+B(0,r))=\mathbb{A}_{m}M_{H_{m}}\dots M_{H_{1}}K+B(0,r),$

that is $(K_{r})_{m}=K_{m}+B(0,r)=(K_{m})_{r}$ and instead of $L_{1}$ and
$L_{2}$ we have the limits $L_{1}+B(0,r)$ and $L_{2}+B(0,r)$. Notice that
$L_{1}\neq L_{2}$ if and only if $L_{1}+B(0,r)\neq L_{2}+B(0,r)$ (see for
example [17], Lemma 3.1.11).

Since $\lambda_{n}(K_{r})>0$, thanks to Lemma 3.1 the sequence of the volumes
$\lambda_{n}(K_{m}+B(0,r))$ is increasing and strictly positive. Moreover it
is bounded, indeed from the compactness of $K$ there exists a ball $B(0,R)$
with $R>0$ such that $K_{r}\subseteq B(0,R)$. For the monotonicity and
symmetry invariance of the Minkowski symmetrization we have that
$K_{m}+B(0,r)\subseteq B(0,R)$ for every $m\in\mathbb{N}$. Then
$\lambda_{n}((K_{m})_{r})$ converges to a certain value $c_{r}>0$.

Since $L_{1}\neq L_{2}$, $\lambda_{n}((L_{1})_{r}\Delta(L_{2})_{r})=\delta>0$.
Fix $0<\varepsilon<\delta/2$, then there exists an index $\nu$ such that
$c_{r}-\lambda_{n}((K_{m})_{r})<\varepsilon$ for every $m\geq\nu$. Consider
for $m>\nu$ the sequence

$J_{m}=\mathbb{A}_{m}\diamondsuit_{H_{m}}\dots\diamondsuit_{H_{\nu+1}}\mathbb{A}_{\nu}^{-1}(K_{\nu})_{r}$
$=\mathbb{A}_{m}\diamondsuit_{H_{m}}\dots\diamondsuit_{H_{\nu+1}}M_{H_{\nu}}\dots
M_{H_{1}}K_{r},$

then thanks to Theorem 2.1, $J_{m}\subseteq(K_{m})_{r}$ and in particular we
have $J_{m_{j}}\subseteq(K_{m_{j}})_{r},J_{m_{l}}\subseteq(K_{m_{l}})_{r}$.
From the hypothesis the sequence $(H_{m})$ is shape-stable in
$\mathcal{K}^{n}$ for $\diamondsuit$, thus there exists $J\in\mathcal{K}^{n}$
such that $J_{m}\to J$. Clearly the same holds for $(J_{m_{j}}),(J_{m_{l}})$.
In particular $J\subseteq(L_{1})_{r}$, $J\subseteq(L_{2})_{r}$ and for Lemma
3.1 $\lambda_{n}(J)\geq\lambda_{n}((K_{\nu})_{r})$. We infer

$\begin{gathered}\lambda_{n}((L_{1})_{r}\Delta(L_{2})_{r})=\lambda_{n}((L_{1})_{r}\setminus(L_{2})_{r})+\lambda_{n}((L_{2})_{r}\setminus(L_{1})_{r})\leq\\\
\lambda_{n}((L_{1})_{r}\setminus J)+\lambda_{n}((L_{2})_{r}\setminus
J)=2c_{r}-2\lambda_{n}(J)\leq
2c_{r}-2\lambda_{n}((K_{\nu})_{r})<2\varepsilon<\delta\end{gathered}$

which is a contradiction, thus $L_{1}=L_{2}$. The same argument can be
repeated for every truncated sequence

$\mathbb{A}_{m}M_{H_{m}}\dots M_{H_{k}}K$

and consequently $(H_{m})$ is shape-stable for the Minkowski symmetrization.

Now we prove that if a sequence is shape-stable in $\mathcal{K}^{n}$ for
Minkowski symmetrizations, then it is shape-stable for
$\spadesuit\in\mathcal{F}$ as well. Consider for $Z\in\mathcal{K}^{n}$ the
sequence

$Z_{m}=\mathbb{A}_{m}\spadesuit_{H_{m}}\dots\spadesuit_{H_{1}}Z.$

Again, by contradiction if $Z_{m}$ does not converge we can find two different
subsequences $Z_{m_{j}},Z_{m_{l}}$ converging respectively to
$W_{1},W_{2}\in\mathcal{K}^{n}$ with $W_{1}\neq W_{2}$.

Thanks to Lemma 3.1 the sequence $\mathcal{W}(Z_{m})$ is non negative and non
increasing, thus $\mathcal{W}(Z_{m})\to b$ for some $b\geq 0$. Then, if
$W_{1}\neq W_{2}$, we have that $\mathcal{W}(\text{conv}(W_{1}\cup W_{2}))>b$.
Notice that the cases $W_{1}\subset W_{2}$ and vice versa are automatically
excluded since $b=\mathcal{W}(W_{1})=\mathcal{W}(W_{2})$ and the mean width is
strictly monotone. Now, for every $\varepsilon>0$ we can find
$\nu\in\mathbb{N}$ such that $\mathcal{W}(Z_{m})-b<\varepsilon$ for every
$m\geq\nu$. Define for every $m\geq\nu$ the sequence

$V_{m}=\mathbb{A}_{m}M_{H_{m}}\dots M_{H_{\nu+1}}\mathbb{A}_{\nu}^{-1}Z_{\nu}$
$=\mathbb{A}_{m}M_{H_{m}}\dots
M_{H_{\nu+1}}\spadesuit_{H_{\nu}}\dots\spadesuit_{H_{1}}Z$

which converges to some $V\in\mathcal{K}^{n}$ because $(H_{m})$ is shape-
stable for Minkowski symmetrization. Minkowski symmetrization preserves the
mean width, thus $\mathcal{W}(V)=\mathcal{W}(Z_{\nu})$. Moreover, thanks to
Theorem 2.1 we have that $W_{1},W_{2}\subseteq V$ and being $V$ convex it
holds $\text{conv}(W_{1}\cup W_{2})\subseteq V$, thus
$\mathcal{W}(V)\geq\mathcal{W}(\text{conv}(W_{1}\cup W_{2}))$, then

$\mathcal{W}(\text{conv}(W_{1}\cup
W_{2}))-b\leq\mathcal{W}(V)-b=\mathcal{W}(Z_{\nu})-b<\varepsilon.$

Being $\varepsilon$ arbitrary this inequality contradicts
$W(\text{conv}(W_{1}\cup W_{2}))>b$ and $W_{1}=W_{2}$. Again the same process
can be applied to the truncated sequences, concluding the proof.

∎

Now Theorem 1.4 is just an easy corollary.

###### Proof of Theorem 1.4.

Observe that if $(H_{m})$ is stable then it is shape-stable with
$\mathbb{A}_{m}$ equal to the identity for every $m$. The proof is then a
straightforward application of Theorem 3.4. ∎

A first consequence is the following extension of Theorem 1.2. Notice that the
extension is twofold: The result holds for the whole family $\mathcal{F}$ and
the respective family of objects is $\mathcal{K}^{n}$ instead of
$\mathcal{K}^{n}_{n}$.

######  Theorem 3.5.

Let $\diamondsuit,\spadesuit\in\mathcal{F}$. A sequence $(H_{m})$ of subspaces
is weakly-universal for $\diamondsuit$ in $\mathcal{K}^{n}$ if and only if the
same holds for $\spadesuit$.

###### Proof.

The strategy will be the same of Theorem 3.4, with the advantage of using
Theorem 1.4.

If $(H_{m})$ is weakly-universal for $\diamondsuit$ in $\mathcal{K}^{n}$, then
it is stable in $\mathcal{K}^{n}$. Using Theorem 1.4, this implies that
$(H_{m})$ is stable for Minkowski symmetrization, thus we only need to prove
that for every $K\in\mathcal{K}^{n}$ the limit $L$ of corresponding
symmetrization process

$K_{m}=M_{H_{m}}\dots M_{H_{1}}K$

is a ball. Again the sequence $\lambda_{n}(K_{m})$ is bounded and increasing,
thus it converges to a certain $c\geq 0$. Through the same argument employed
in Theorem 3.4 we can suppose $c>0$, i.e. considering $K+B(0,r)$ instead of
$K$ for $r>0$ arbitrary small.

Since $(H_{m})$ is weakly-universal for $\diamondsuit$, for every
$\nu\in\mathbb{N}$ we have that the sequence

$\diamondsuit_{H_{m}}\dots\diamondsuit_{H_{\nu+1}}M_{H_{\nu}}\dots M_{H_{1}}K$

converges to a ball $B_{\nu}$ such that
$\lambda_{n}(B_{\nu})\geq\lambda_{n}(K_{\nu})$ thanks to Lemma 3.1 and
$B_{\nu}\subseteq L$ for every $\nu$ thanks to Theorem 2.1. Since
$\lambda_{n}(K_{m})\to c$ increasingly, we have that for every $\varepsilon>0$
exists $\nu\in\mathbb{N}$ such that $\lambda_{n}(L\Delta
B_{\nu})<\varepsilon$, thus $L$ is a ball.

Suppose now that $(H_{m})$ is weakly universal for Minkowski symmetrization in
$\mathcal{K}^{n}$, then clearly $(H_{m})$ is stable for $\spadesuit$. Consider
then for $Z\in\mathcal{K}^{n}$ the limit $W$ of the sequence

$Z_{m}=\spadesuit_{H_{m}}\dots\spadesuit_{H_{1}}Z.$

Again $\mathcal{W}(Z_{m})$ is a non negative and non increasing sequence, thus
it converges to a value $b\geq 0$.

$(H_{m})$ is weakly-universal for Minkowski symmetrization, thus for every
$\nu\in\mathbb{N}$ we have a ball $B_{\nu}$ as the limit of the sequence

$M_{H_{m}}\dots M_{H_{\nu+1}}\spadesuit_{H_{\nu}}\dots\spadesuit_{H_{1}}Z.$

Then $\mathcal{W}(B_{\nu})=\mathcal{W}(Z_{\nu})$ thanks to the properties of
Minkowski symmetrization and for every $\nu$ Theorem 2.1 gives $W\subseteq
B_{\nu}$. Since $\mathcal{W}(B_{\nu})$ converges decreasingly to
$\mathcal{W}(W)$, we have that $W$ must be a ball. ∎

Theorem 3.4 gives us the possibility to extend many known results for Steiner
symmetrization to all the $\diamondsuit$ satisfying (1,3,4), in particular the
Minkowski symmetrization through hyperplanes. For example we immediately have
the following generalization of Theorem 2.2.

######  Corollary 3.6.

Let $(H_{m})$ be a sequence of hyperplanes and the corresponding normals
$(u_{m})\subset\mathbb{S}^{n-1}$. Consider moreover
$\diamondsuit\in\mathcal{F}$ and the angles $(\alpha_{m})$ such that
$u_{m}\cdot u_{m-1}=\cos\alpha_{m}$. If
$\sum_{m\in\mathbb{N}}\alpha_{m}^{2}<+\infty$, then there exist rotations
$(R_{m})$ such that for every non empty convex compact set
$K\subset\mathbb{R}^{n}$ the sequence

$K_{m}:=R_{m}\diamondsuit_{H_{m}}\dots\diamondsuit_{H_{1}}K$

converges in Hausdorff distance to a set $L\in\mathcal{K}^{n}$.

For the Minkowski symmetrization with respect to hyperplanes we have a
stronger result. First we need the following convergence criterion from [18]
(Theorem 1.2).

######  Theorem 3.7.

Consider $K\in\mathcal{K}^{n}$ and a sequence of isometries
$(\mathbb{A}_{m})$. If the sequence

$K_{m}=\frac{1}{m}\sum_{j=1}^{m}\mathbb{A}_{j}K$

converges, then the same happens for every compact set $C\in\mathcal{C}^{n}$
such that conv$(C)=K$. Moreover, the two sequences converge to the same limit.

Together with Theorem 1.4 this implies the following.

######  Corollary 3.8.

If $(H_{m})$ is a shape-stable sequence of hyperplanes for Steiner
symmetrization on $\mathcal{C}^{n}$, then it is shape-stable on
$\mathcal{C}^{n}$ for Minkowski symmetrization. In particular Theorem 2.2
holds for Minkowski symmetrization as well.

###### Proof.

First observe that being $\mathcal{K}^{n}$ closed in $\mathcal{C}^{n}$, then
$(H_{m})$ is obviously shape-stable for Steiner symmetrization on
$\mathcal{K}^{n}$ and from Theorem 3.4 it is shape-stable for Minkowski
symmetrization on $\mathcal{K}^{n}$.

Now, in order to conclude the proof we just have to express the shape-stable
sequence as a sequence of means of isometries so that we can apply Theorem
3.7. This is clear if we write explicitly the sequence. Indeed, for every
$C\in\mathcal{C}^{n}$

$C_{1}=\mathbb{A}_{1}M_{H_{1}}C=\mathbb{A}_{1}\left(\frac{C+R_{H_{1}}C_{1}}{2}\right)=\frac{\mathbb{A}_{1}C+\mathbb{A}_{1}R_{H_{1}}C}{2}.$

Iterating this process every $C_{m}$ is a Minkowski mean of $2^{m}$ isometries
of $C$.

Theorem 2.2 provides shape-stable sequences for Steiner symmetrization in
$\mathcal{C}^{n}$, thus the same sequences are shape-stable in
$\mathcal{C}^{n}$ for Minkowski symmetrization. ∎

We conclude this section with the proof of Theorem 1.5.

###### Proof of Theorem 1.5.

In the hypothesis of the theorem we can apply Corollary 3.6. The rotations
$R_{m}$ in such statement (for the details on their construction see the proof
of Theorem 2.2 in [1], in this paper Theorem 2.2) correspond to the
composition of the $m$ planar rotations $\mathbb{A}_{m}$ of $\alpha_{m}$
degrees as $R_{m}=\mathbb{A}_{m}\dots\mathbb{A}_{1}$, where every
$\mathbb{A}_{m}$ is such that $\mathbb{A}_{m}R_{m-1}u_{m}=e_{1}$ and it fixes
the subspace $u_{m}^{\perp}\cap e_{1}^{\perp}$.

We now show that $(R_{m})$ is a Cauchy sequence on the space of endomorphisms
of $\mathbb{R}^{n}$ with the classical sup norm

$\norm{\mathbb{A}}:=\sup_{z\in\mathbb{S}^{n-1}}\norm{\mathbb{A}z},$

thus it converges to a certain rotation $R$. From the properties of the norm
and the triangular inequality

$\begin{gathered}\norm{R_{m+k}-R_{m}}=\norm{\mathbb{A}_{m+k}\dots\mathbb{A}_{m+1}R_{m}-R_{m}}=\norm{R_{m}}\norm{\mathbb{A}_{m+k}\dots\mathbb{A}_{m+1}-Id}=\\\
\norm{\mathbb{A}_{m+k}\dots\mathbb{A}_{m+1}-Id}\leq\norm{\mathbb{A}_{m+k}\dots\mathbb{A}_{m+1}-\mathbb{A}_{m+1}}+\norm{\mathbb{A}_{m+1}-Id}\leq\\\
\norm{\mathbb{A}_{m+1}}\norm{\mathbb{A}_{m+k}\dots\mathbb{A}_{m+2}-Id}+2\sin(\absolutevalue{\alpha_{m+1}}/2)\leq\dots\leq
2\sum_{j=m+1}^{m+k}\sin(\absolutevalue{\alpha_{j}}/2).\end{gathered}$

From the hypothesis the series $(\absolutevalue{\alpha_{m}})$ converges,
proving the claim.

From Corollary 3.6 the sequence of sets

$R_{m}\diamondsuit_{H_{m}}\dots\diamondsuit_{H_{1}}K$

converges for every $K\in\mathcal{K}^{n}$ to a certain set $L$, then it makes
sense to consider the set $R^{-1}L$. We then have the following estimates

$d_{\mathcal{H}}(\diamondsuit_{H_{m}}\dots\diamondsuit_{H_{1}}K,R^{-1}L)\leq
d_{\mathcal{H}}(\diamondsuit_{H_{m}}\dots\diamondsuit_{H_{1}}K,R_{m}^{-1}L)+d_{\mathcal{H}}(R_{m}^{-1}L,R^{-1}L).$

Thanks to the invariance with respect to isometries of the Hausdorff distance

$d_{\mathcal{H}}(R_{m}\diamondsuit_{H_{m}}\dots\diamondsuit_{H_{1}}K,L)=d_{\mathcal{H}}(\diamondsuit_{H_{m}}\dots\diamondsuit_{H_{1}}K,R_{m}^{-1}L)$

and thus from the convergence of
$R_{m}\diamondsuit_{H_{m}}\dots\diamondsuit_{H_{1}}K$ and $R_{m}$ we infer
that $\diamondsuit_{H_{m}}\dots\diamondsuit_{H_{1}}K$ converges to $R^{-1}L$,
concluding the proof.

∎

## 4 Counterexamples to convergence

As we have seen Theorem 3.4 lets us extend Theorem 2.1 to all the
symmetrizations $\diamondsuit\in\mathcal{F}$. The latter result arises from
the study of a peculiar counterexample which now we briefly show. It can be
found in different versions in [5],[6] and [1]. In the following examples the
vectors $\\{e_{1},e_{2}\\}$ are intended as the standard orthonormal basis in
$\mathbb{R}^{2}$.

######  Example 4.1.

Consider a sequence of angles $(\alpha_{m})\subset(0,\pi/2)$ such that

$\sum_{m\in\mathbb{N}}\alpha_{m}=+\infty,\qquad\sum_{m\in\mathbb{N}}\alpha_{m}^{2}<+\infty,$
(4.1)

and take the sequence of directions in $\mathbb{R}^{2}$ given by
$u_{m}:=(\cos\beta_{m},\sin\beta_{m})$ where

$\beta_{m}:=\sum_{j=1}^{m}\alpha_{j}$

with corresponding orthogonal lines $H_{m}=u_{m}^{\perp}$.

Let $0<\gamma:=\prod_{m\in\mathbb{N}}\cos\alpha_{m}$ (which converges because
of the second condition in (4.1)), we consider a compact set
$K\subset\mathbb{R}^{2}$ with area $0<|K|<\pi(\gamma/2)^{2}$ and containing a
vertical unitary segment $\ell$ centered in the origin. We prove now that the
sequence

$K_{m}:=S_{H_{m}}\dots S_{H_{1}}K$

does not converge. Consider indeed the sequence of segments

$\ell_{m}:=P_{H_{m}}K_{m-1},$

where the length of $\ell_{m}$ converges to $\gamma>0$ and each of them is
clearly aligned with $H_{m}$. The sequence of directions $(u_{m})$ is dense in
$S^{1}$, thanks to (4.1), and it does not converge. The same holds for the
perpendicular sequence of lines $(H_{m})$. Thus for every
$\nu\in\mathbb{S}^{1}$ we can find a subsequence $H_{m_{k}}^{\perp}$ such that
its corresponding direction converges to $\nu$. Similarly $\ell_{m_{k}}$
converges to a segment of length $\gamma>0$ aligned to $\nu$.

Now, if $K_{m}$ converges, for the monotonicity of Steiner symmetrization it
must contain all these subsequences of diameters, and consequently a ball $B$
of diameter $\gamma$ centered in the origin. But we supposed
$\lambda_{2}(K)<\pi(\gamma/2)^{2}$, thus $K_{m}$ cannot converge.

The peculiarity of the sequence involved in this example is that the
corresponding directions are dense in $\mathbb{S}^{1}$, which could seem a
reasonable sufficient condition for convergence to a ball. As it was showed
this is definitely not the case, even though in [5] it was proved for compact
convex sets that a dense sequence of hyperplanes can be reordered to obtain a
universal sequence. This was generalized in [20] to generic compact sets.

In [4] instead it is proved a characterization of the symmetry that a convex
compact body needs in order to be a ball. The form we present includes the
statements from Theorem 3.2 for one dimensional subspaces.

######  Theorem 4.1 (Bianchi, Gardner and Gronchi).

Let $H_{j}\in\mathcal{G}(n,1),j=1,\dots,n$, be such that
(i) at least two of them form an angle that is an irrational multiple of
$\pi$,
(ii) $H_{1}+\dots+H_{n}=\mathbb{R}^{n}$ and
(iii)${H_{1},\dots,H_{n}}$ cannot be partitioned into two mutually orthogonal
non empty subsets.

If $E\subseteq\mathbb{S}^{n-1}$ is nonempty, closed and such that
$R_{H_{j}}E=E,j=1,...,n$, then $E=\mathbb{S}^{n-1}$.

Hence, if $K\in\mathcal{K}^{n}_{n}$ satisfies $R_{H_{j}}K=K$ for
$j=1,\dots,n$, then $K$ is a ball centered in the origin.

We can use this theorem to find sequences of lines such that the corresponding
symmetrization process, if it converges, goes to a ball. Indeed, consider a
sequence $(v_{m})\subset\mathbb{S}^{n-1}$ with $n$ accumulation points
generating a family of lines ${H_{1},\dots,H_{n}}$ as in the statement of
Theorem 4.1. Consider indeed a sequence $(K_{m})$ of convex bodies such that
every $K_{m}$ is symmetric with respect to $v_{m}^{\perp}$. Then, if the
sequence converges, the limit must necessarily be a ball. We can use this fact
to provide a new kind of counterexample.

######  Example 4.2.

Consider in $\mathbb{R}^{2}$ the two directions
$w_{1}=(1,0),w_{2}=(\cos\alpha,\sin\alpha)$ such that $\alpha>0$ is an
irrational multiple of $\pi$. We consider a sequence
$(\gamma_{m})\subset[0,\alpha]$ such that
$(\alpha_{m}):=(\absolutevalue{\gamma_{m+1}-\gamma_{m}})$ is as in (4.1).
Moreover we want $\alpha$ and $0$ to be accumulation points of $(\gamma_{m})$.

Consider the sequence of lines
$H_{m}=span\\{(\cos\gamma_{m},\sin\gamma_{m})\\}$, then the corresponding
directions go back and forth between $w_{1}$ and $w_{2}$. Observe that
$(H_{m})$ has two accumulation points aligned with $w_{1}$ and $w_{2}$.

Let $K$ be a compact body centered in the origin with a unitary diameter
parallel to $w_{1}$ and consider the sequence of symmetrals

$K_{m}=S_{H_{m}}\dots S_{H_{1}}K.$

As in Example 4.1 we can consider a sequence of segments

$\ell_{m}=K_{m}\cap H_{m}$

such that $\lambda_{1}(\ell_{m+1})\geq\lambda_{1}(\ell_{m})\cos\alpha_{m+1}$,
thus $\lambda_{1}(\ell_{m})$ converges to a certain value $\gamma>0$ and in
particular the two limits of the converging subsequences of $(\ell_{m})$
respectively aligned with $w_{1}$ and $w_{2}$ will have length greater than
$\gamma$.

Using Theorem 4.1, if $K_{m}$ converges then the limit must be a ball. If we
choose $\lambda_{2}(K)<\pi(\gamma/2)^{2}$, such ball should contain a diameter
of length $\gamma$, which is not possible, thus $K_{m}$ cannot converge.

We conclude proving that Example 4.1 can be generalized for other
symmetrizations, again thanks to Theorem 2.1.

######  Example 4.3.

Consider a set $K\in\mathcal{K}_{2}^{2}$ such that it contains a unitary
horizontal segment and with mean width $1/2\pi<\mathcal{W}(K)<\gamma$, where
$\gamma$ is as in Example 4.1. In the hypothesis of Theorem 2.1, for
$\diamondsuit\in\mathcal{F}$ we have that

$S_{U_{m}}...S_{U_{1}}K\subseteq\diamondsuit_{U_{m}}...\diamondsuit_{U_{1}}K\subseteq
M_{U_{m}}...M_{U_{1}}K,$

again $U_{j}:=\text{span}(u_{j})$, and we used Steiner symmetrization because
it is equivalent to fiber symmetrization relative to a hyperplane, which is
our case working on $\mathbb{R}^{2}$.

In this way we can exploit the first counterexample and the inclusion chain of
Theorem 2.1 to guarantee that, if a limit exists for
$\diamondsuit_{U_{m}}...\diamondsuit_{U_{1}}K$ and $M_{U_{m}}...M_{U_{1}}K$,
proceeding as before it must contain a ball of diameter $\gamma$, therefore
this limit must have mean width greater than $\gamma$. In particular this
holds for the sequence of Minkowski symmetrals. But Minkowski symmetrization
preserves mean width, which we supposed to be less than $\gamma$. This is a
contradiction, thus there cannot be a limit.

## Acknowledgements

The author would like to thank Gabriele Bianchi and Paolo Gronchi for the many
discussions and insightful conversations during the preparation of this work.

## References

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# Firsthand Opiates Abuse on Social Media: Monitoring Geospatial Patterns of
Interest Through a Digital Cohort

Duilio Balsamo  University of Turin & ISI FoundationTurinItaly
<EMAIL_ADDRESS>,  Paolo Bajardi  ISI FoundationVia Chisola
5TurinItaly10126<EMAIL_ADDRESS>and  André Panisson  ISI FoundationVia
Chisola 5TurinItaly10126<EMAIL_ADDRESS>
(2019)

###### Abstract.

In the last decade drug overdose deaths reached staggering proportions in the
US. Besides the raw yearly deaths count that is worrisome per se, an alarming
picture comes from the steep acceleration of such rate that increased by 21%
from 2015 to 2016. While traditional public health surveillance suffers from
its own biases and limitations, digital epidemiology offers a new lens to
extract signals from Web and Social Media that might be complementary to
official statistics. In this paper we present a computational approach to
identify a digital cohort that might provide an updated and complementary view
on the opioid crisis. We introduce an information retrieval algorithm suitable
to identify relevant subspaces of discussion on social media, for mining data
from users showing explicit interest in discussions about opioid consumption
in Reddit. Moreover, despite the pseudonymous nature of the user base, almost
1.5 million users were geolocated at the US state level, resembling the census
population distribution with a good agreement. A measure of prevalence of
interest in opiate consumption has been estimated at the state level,
producing a novel indicator with information that is not entirely encoded in
the standard surveillance. Finally, we further provide a domain specific
vocabulary containing informal lexicon and street nomenclature extracted by
user-generated content that can be used by researchers and practitioners to
implement novel digital public health surveillance methodologies for
supporting policy makers in fighting the opioid epidemic.

Online disease monitoring, Opioid epidemic, Reddit

††journalyear: 2019††copyright: iw3c2w3††conference: Proceedings of the 2019
World Wide Web Conference; May 13–17, 2019; San Francisco, CA, USA††booktitle:
Proceedings of the 2019 World Wide Web Conference (WWW ’19), May 13–17, 2019,
San Francisco, CA, USA††doi: 10.1145/3308558.3313634††isbn:
978-1-4503-6674-8/19/05††ccs: Information systems Information retrieval††ccs:
Applied computing Health informatics††ccs: Human-centered computing Social
media

##  1\. Introduction

The opioid crisis emerged in the US from the interplay of several determinants
and evolved over time through three major phases: from the abuse of drug
prescriptions of late 80’s (Leung et al., 2017), to heroin injecting users
soaring in 2010 (Michel and Loscalzo, 2015) and the very recent rise of
synthetic opioids flooding the drug market (Kolodny et al., 2015). The rate of
overdose deaths involving opioids shows heterogeneous geographical patterns
ranging from 4.9 per 100,000 inhabitants in Texas to 43.4 in West Virginia in
2016 (Seth et al., 2018). Unfortunately, despite the effort of national
agencies in monitoring the phenomenon, state-level estimates suffer biases due
to unequal coverage of the surveillance system as well as intrinsic biases in
counting overdose deaths that lead to low specificity of drugs involved
(Hedegaard et al., 2014; Ruhm, 2017; Landen et al., 2003). Besides drug
overdose maps, opioid prescription maps are provided on yearly basis by the
Centers for Disease Control and Prevention (CDC), showing heterogeneous
patterns also due to different prescribing policies implemented by individual
states. In this perspective, the gold standard represented by official
surveillance statistics has to be carefully considered in the light of the
known biases of the reported numbers and estimates. A complementary way to
investigate these phenomena is by field studies, through ethnographic
approaches and structured interviews: sadly these methods require an
exceptional effort in order to gather insights from firsthand users (Mars et
al., 2014) or subjects under opioid substitution maintenance treatment (Sordo
et al., 2017), and are often limited to small sample sizes ranging from tens
to hundreds of individuals. In this context, a digital epidemiology approach
(Salathé et al., 2012) aimed at integrating and complementing existing
knowledge about the opioid crisis gathering information with a bottom-up
unsolicited approach might be extremely valuable.

In recent years, Social Media changed the way drug users share information
about opiates, give online warnings and avoid potentially toxic drug batches
(III, 2017), pointing at “Reddit: the front page of Internet” (Reddit, 2018)
as a promising digital source of information. Reddit is a social content
aggregation website, the 5th most popular website in the US (Alexa.com, Fall
2018), on which users can post, comment and vote content. It is structured in
cross-referenced yet independent sub-communities (i.e. subreddits), centered
around a variety of topics ranging from science to drugs and porn (Medvedev et
al., 2018). Reddit is constantly growing, with a total of almost 800 million
comments in the year 2016 and almost a billion comments in the year 2017.
Given the ease to register with a “throwaway” account, Reddit is often used to
discuss topics otherwise considered socially unacceptable or unsuitable for
the mainstream; users can actively engage with others, talking uninhibitedly
about personal experiences (Manikonda et al., 2018), receiving back social
support and even life-saving tips from sub-communities of peers (III, 2017).
However, navigating such massive platforms and finding areas of specific
interest is usually cumbersome since topics are self-organized bottom-up
through users’ interactions, leaving the users to find relevant topics by word
of mouth or using a basic search feature.

The main contributions of this work are summarized as follows:

  * •

Design a general purpose information retrieval algorithm able to identify
regions of interest when conducting epidemiological surveillance and
monitoring on social media

  * •

Provide an open domain specific vocabulary related to opiates discussions

  * •

Demonstrate how information disclosed by Reddit users can be used to estimate
their geographical location

  * •

Identify a novel digital cohort of individuals addressing in a pseudonymous
fashion health related topics

  * •

Provide prevalence maps of interest in opiates consumption

##  2\. Related work

Digital epidemiology (Salathé et al., 2012), also referred to as digital
disease detection (Brownstein et al., 2009a) and infodemiology (Eysenbach,
2009), broadly includes the use of the Internet and digital technologies for
collecting and processing health information, both in aggregate form for
public health surveillance or from individuals for personal health monitoring.
Data from a so called digital cohort might be collected with active
participation of individuals, as in the case of participatory epidemiology
(Freifeld et al., 2010), but might also be collected passively, e.g. from
Social Media, as a byproduct of platforms designed for different purposes.
Participatory systems have been implemented through the use of Web platforms
(Paolotti et al., 2014) and signals collected from such systems have been
shown to be useful for epidemic forecasting (Zhang et al., 2017). Data
collected from the Web and Social Media have also shown to be useful for
monitoring different infectious diseases (Kass-Hout and Alhinnawi, 2013;
Chunara et al., 2012; Brownstein et al., 2009b; Brownstein et al., 2017;
Sharma et al., 2017).

Reddit has already proven to be suitable for a variety of research purposes,
ranging from the study of user engagement and interactions between highly
related communities (Tan and Lee, 2015; Hessel et al., 2016) to post-election
political analyses (Barthel, 2016). Also, it has been useful to study the
impact of linguistic differences in news titles (Horne and Adali, 2017) and to
explore recent web-related issues such as hate speech (Saleem et al., 2017) or
cyberbullying (Rakib and Soon, 2018) as well as health related issues like
mental illness (Choudhury and De, 2014), also providing insights about the
opioid epidemics (Park and Conway, 2017). It is worth to notice that since the
platform is collectively generated by the users as in most Social Media, there
is not a blueprint of Reddit to perimeter the area under study, and only few
attempts tried to overcome this issue extracting meaningful maps or suitable
embedding spaces (Olson and Neal, 2015; Martin, 2017).

We approach the problem of selecting subreddits that are relevant for a
specific topic as an information retrieval (IR) problem, where it is possible
to retrieve topic-specific documents by expressing a limited set of known
keywords. Language models (Ponte and Croft, 1998; Schütze et al., 2008; Croft
and Lafferty, 2013) tackle this problem using a probabilistic approach with
the idea that words that are relevant for a given topic would be more likely
to appear in a relevant document. While language modeling is not a common
approach for ranking documents collected from Social Media due to the inherent
sparsity of the documents – e.g. for Twitter, more elaborated IR approaches
are needed to resolve the sparsity of short texts for tweets (Naveed et al.,
2011) – subreddits, on the other hand, can be seen as very rich documents from
which topic-specific word distributions can be built.

The set of keywords expressed by the user may not include some yet unknown
terms that are relevant to the topic but very specific to the language models
of the documents, pointing the necessity of query expansion techniques.
Relevance feedback (Rocchio, 1971) and pseudo-relevance feedback (Buckley et
al., 1995) are common approaches for query expansion; many of these approaches
use human judgment to manually select a subset of the top retrieved documents,
and use them to expand the query. More recently, word embeddings (Kuzi et al.,
2016) have been used to expand the query with terms that are semantically
similar to the query terms. Techniques that are based on language modeling
might incorporate term proximity information (Ermakova et al., 2016) to
address the automatic query expansion problem, or use an approach based on
Information Theory, exploiting the Kullback-Leibler distance for query
reweighing (Carpineto et al., 2001) and for training local word embeddings
(Diaz et al., 2016).

##  3\. Dataset and data preparation

A dataset 111https://files.pushshift.io/reddit/ containing the list of all
submissions and comments published in Reddit since 2007 is publicly available
online (Baumgartner, 2015) and is maintained monthly by adding recent entries.
The dataset is not 100% complete and it contains some gaps (Gaffney and
Matias, 2018), but these are very small for the years 2016 and 2017 (around 1%
missing data per month). We use the union of all submissions and comments from
years 2016 and 2017, for a total of 1,980,497,553 entries. Only subreddits
with at least 100 entries were selected, resulting in a set of 1,973,863,886
entries with 74,810 distinct subreddits and 15,747,502 distinct users. The
text of each entry is parsed and tagged using the spaCy NLP library
222https://spacy.io/ v1.9.0. For the part-of-speech tagging, a greedy averaged
perceptron model was used (Honnibal and Johnson, 2015). Finally, lemmatization
is applied to each POS tag; the English lemmatization data is taken from
WordNet (Miller, 1995) and lookup tables are taken from Lexiconista
333http://www.lexiconista.com/datasets/lemmatization/. After all terms are
lemmatized, we select those that appear at least 100 times in the corpus,
resulting in a vocabulary of 762,746 lemmas.

##  4\. identifying relevant documents via document ranking and query
expansion

As discussed in Section 2, the wealth of information contained in Reddit data
is not readily available and has to be thoroughly mined. In this section we
describe an iterative methodology of semi-automatic retrieval of documents in
heterogeneous corpora, in which human intervention is as little as possible.
It is worth to stress that the approach is general and fully unsupervised.
However, we added a human-in-the-loop to include domain expert knowledge in
the process and reach better results. On the other hand, a domain expert alone
without the aid of the algorithmic pipeline for document ranking and query
expansion would have been hopeless in navigating the Reddit world by hand. The
steps are summarized in Algorithm 1. We start with a small set of keywords $Q$
provided by the user, or query in the following. At each iteration, we select
documents which are both relevant to the query and informative, and enrich the
query terms set until we arrive to a stable list of documents and query terms.
While this methodology works well on Reddit in the domain of topics related to
the opioid epidemics (see Section 4.1), it is also sufficiently general to be
used for other information retrieval tasks and might be valuable for different
epidemiological research questions.

Input: Corpus $C$, query $Q$

Parameters : $n$, $m$, $\alpha$

1 Initialize the vocabulary V;

2 foreach  _word $w\in\textbf{V}$_ do

3 calculate $p_{C}(w)$;

4 foreach  _document $d\in C$_ do

5 calculate $p_{d}(w)$;

6

7

8$Q_{new}\leftarrow Q$;

9 $K_{new}\leftarrow\varnothing$;

10 repeat

11 $Q\leftarrow Q_{new}$;

12 $K\leftarrow K_{new}$;

13 $R_{d}$ $\leftarrow$ Rank documents using $\mathrm{score}(d\mid Q,C)$ (Eq.
2);

14 $K_{new}$ $\leftarrow$ top $n$ documents in $R_{d}$;

15 $R_{w}$ $\leftarrow$ Rank terms using $\mathrm{score}(w\mid K_{new})$ (Eq.
3);

16 $Q_{candidate}$ $\leftarrow$ top $m$ terms from $R_{w}$;

17 $Q_{new}$ $\leftarrow$ manual selection of terms in $Q\cup Q_{candidate}$;

18

19until  _$Q=Q_{new}$ and $K=K_{new}$_;

Output: $R_{d}$, $R_{w}$

Algorithm 1 IR steps for document ranking

First, we create a general vocabulary V by collecting terms from the entire
corpus in a bag-of-words fashion. We compute the probability of occurrence of
a term $w$ in the entire corpus $C$ as the ratio
$p_{C}(w)=f_{C}(w)/\sum\limits_{w}f_{C}(w)$ between its raw count $f_{C}(w)$
in the corpus and the total number of words in the corpus. Let us also define
the regularized marginal probability of occurrence of term $w$ in a document
$d$ as

(1) $p_{d}(w)=\frac{f_{d}(w)}{\sum_{w}f_{d}(w)+\alpha}+p_{C}(w).$

where $f_{d}(w)$ is the count of $w$ in $d$. In case of corpora with very
heterogeneous document sizes, the regularization term $\alpha$ is added to
control “the size” of the language model of the documents (small documents
will result in small marginal probabilities). In our experiments, we use
$\alpha=10^{4}$, so documents with total number of words lower than $10^{4}$
have “flattened” probabilities in their language model. Adding $p_{C}(w)$ to
the marginal probability of $w$ reduces the impact of words that are rare or
not present in a document, and only words that are more likely to appear in
the document will impact the document ranking. The use of these two
regularization terms ($\alpha$ and $p_{C}(w)$) are effective in low-count
scenarios and have a small impact for words with high probability in the
document or in case of documents with large language model.

With the intuition that a document will result to be relevant in the context
of the query if it contains query terms more likely to appear in the document
than in the general corpus, we evaluate

(2) $\mathrm{score}(d\mid Q,C)=KLD_{Q}(p_{d},p_{C})=\sum_{w\in
Q}p_{d}(w)log\frac{p_{d}(w)}{p_{C}(w)}$

which is the total contribution of the query terms in the Kullback-Leibler
divergence between the document and the whole corpus $C$.
We consider the top $n$ documents ranked by relevance as measured by Eq. (2)
as the set of relevant documents $K$. Once $n$ is chosen (and thus $K$ is
obtained), in order to enrich the query terms $Q$ we assign a score to each
term based on the logarithm of the likelihood ratio

(3) $\mathrm{score}(w\mid K)=\sum_{k\in
K}log\frac{p_{k}(w)+p_{C}(w)}{p_{C}(w)}.$

Whenever the the maximum likelihood estimate $p_{k}(w)$ of a term in a
relevant document is higher than its estimate in the general context
$p_{C}(w)$ – i.e. the term is more informative in the document than in the
general context – the contribution to the score will be high and positive.
Conversely, whenever a term is less likely to appear in the document than in
general context, its contribution to the score will be smaller, highlighting
that the term is common or not relevant to the context. Considering the top
$m$ terms ranked by Eq. (3), a subset of previously unknown relevant terms is
added to $Q$ with the supervision of an expert. With the newly enriched set of
query terms, the entire pipeline can be iteratively evaluated until
convergence, reached if no new documents are added to the top $n$ documents
and no new relevant terms among the top $m$ can be added to $Q$. These steps
are summarized in Algorithm 1, and the document ranking resulting from the
last iteration of the algorithm is used to select the most relevant documents.

Figure 1. Opiates subreddits (top): individual subreddits are shown in a
coordinate space of the number of words in the subreddit and the final ranking
score. The subreddits above the dashed line are those selected as $K$ opiates
related subreddits. Opiates vocabulary (bottom): Top 2,000 terms sorted by
rank probability of term in the set $K$. Terms above the dashed line were
selected as query term candidates in the last step of the subreddit retrieval
algorithm.

###  4.1. Opioid related subreddits

Our assumption is that authors who post content in a subreddit related to a
particular topic are interested in that topic. Therefore we consider all
authors participating in threads on subreddits related to opioid consumption
as those interested in the topic, and we discover such subreddits by applying
the algorithm described in Section 4.1. After data preparation steps described
in Section 3, starting from a list of opiates related keywords $q=$ [fentanyl,
oxycodone, suboxone, morphine] we extract the top $n=10$ subreddits:
[suboxone, fentanyl, Opiatewithdrawal, TarkovTrading, heroinaddiction,
ChronicPainPlayhouse, OpiatesRecovery, opiates, Methadone, PoppyTea]. We then
proceed with the iterative procedure of query enrichment and document ranking,
considering as relevant terms only opioid drug names, i.e. chemical names
(e.g. oxycodone), brand names (e.g. Percocet) and street slang (e.g. percs),
disregarding drugs that might be abused together with opiates (like
benzodiazepines) but are not in the opiates domain. The final set of opioid
related subreddits used in this paper is $K=$ [fentanyl, suboxone,
Opiatewithdrawal, Methadone, opiates, pillhead, lean, ChronicPainPlayhouse,
heroin-addiction, OpiatesRecovery]. All the 37,009 users who posted on such
subreddits in 2016 or 2017 are then considered as interested in opiates.
Figure 1 (top) shows the final ranking score for the subreddits, plotted
against their size in terms of number of words.

###  4.2. Opioid specific vocabulary

When applied to Reddit, the query expansion approach is particularly useful in
revealing how a topic is discussed on subreddits. Given the large user base of
Reddit and the tendency of the users in employing slang and street names
alongside proper drug names, this method is very helpful in acknowledging
alternative names of drugs, like sub for suboxone and bth for black tar
heroin.

As a byproduct of the proposed methodology applied to the opiates domain, we
extracted a topic-specific vocabulary by weighting each term of the vocabulary
with Eq. 3. Very specific opioid-related terms (i.e. with high probability in
opiates subreddits and low probability in the whole corpus) have large
positive values of $score(w\mid K)$, as shown in Figure 1 (bottom), while
stop-words and common terms have small score values. A total of 2,616 terms
out of the original 762,746 (0.3%) have score higher than 1. The full list of
vocabulary terms ranked by score is available for research purposes
444https://github.com/ISIFoundation/WWW19OpiatesAbuseSocialMedia.

##  5\. Geolocating users on Reddit

Reddit does not provide any explicit information about users’ location,
therefore we apply three methodologies to assign a location to users. Finally,
we merge the mappings in a single user-location matching.

1\. Self reporting via regular expression: Reddit users often declare
geographical information about themselves in submission or comment texts. We
selected all texts containing the expression ‘I live in’ (3,337,850 instances
in 2016 and 2017) and extracted candidate expressions from the text that
follows, to identify which ones represented US states and cities. We started
with a set of US cities from the GeoNames database 555http://www.geonames.org/
with population higher than 20k, and selected only the candidate expressions
that included both the city name and the state (e.g. ‘Newark, California’ or
‘Newark, CA’) to avoid confusion with cities with same name (e.g. ‘Newark, New
Jersey’). Once the US state for these expressions were assigned and removed
from the candidate expressions, we proceeded to all US cities with population
higher than 200k, selecting expressions with the name of the city and their
variants (e.g. ‘New York’, ‘Big Apple’). After assigning the corresponding US
state for these expressions and removing them from the candidates, we
proceeded to select the expressions with a state name on it (e.g. ‘Alabama’,
‘California’). Among the initial set of candidate expressions, 886,919 (27%)
had a state associated to it. By removing inconsistent self reporting (13,374
users who reported more than one US state) we geolocated 378,898 distinct
users.

2\. Self reporting via user flairs: In Reddit, user flairs are attributes
(usually selected by the users) that are attached to their submissions or
comments in a specific subreddit. In some subreddits flairs might be limited
to a set of geographical locations (countries, states, cities and city
neighborhoods), meaning that users should identify themselves with one of
these locations. A user selecting a location flair is therefore considered
equivalent to a user self reporting its location. We mapped the users
participating in subreddits with location flairs referring to US states to
their flaired positions. Using this approach, we mapped 206,125 users to the
51 US states (including District of Columbia) by selecting the most common
among the position flairs expressed by a user.

3\. Posting on subreddits specific to locations: Reddit includes subreddits
discussing topics specific to geographical locations (e.g. r/Alabama or
r/Seattle). The subreddit r/LocationReddits keeps a curated list of these
local subreddits. We collected from the page corresponding to North
America666https://www.reddit.com/r/LocationReddits/wiki/faq/northamerica the
mappings of 2,844 subreddits to 51 US states. By assuming that a user who
posts comments in one of these subreddits is likely to live that location, we
estimated the position of 1,198,096 authors.

After retrieving US states positions using the three methods above, we found
that about 12% of mapped users expressed multiple locations. In order to
uniquely map authors and states, for location flairs and LocationReddits
sources we assigned each author a unique location, it being the most frequent
among the ones expressed by the author. We discarded authors whose most
frequent location was not unique. This resulted in 194,008 authors retrieved
via location flairs (5.9% loss) and 1,077,516 via LocationReddits (1.4% loss).

Figure 2.  Reddit geolocated population: scatter plot of the number of
geolocated Reddit users and census population.

We evaluated Pearson’s $r$ correlation between the log of the 2000 US Census
population and the log of the population assigned to the same US states using
the three methodologies. Results are in good agreement for all sources, with
$r$= 0.85, 0.91 and 0.86 for respectively User flairs, Regular expression and
LocationReddits, and all $p$-values below 1e-12.

Finally, we merged the information from all three sources in a unique location
for each author. We considered the regular expression technique to be the most
reliable due to its unambiguous self reporting nature, resulting in the
highest correlation with census data. We proceeded in the merging process by
first assigning the authors their regular expression location, if present. If
missing, we assigned them their position from the joint information of
location flairs and LocationReddits by summing the occurrences of locations
expressed in the two sources and verifying the uniqueness of the most frequent
location. Although some approaches have been proposed to geolocate users using
language models (Han et al., 2014), we rely on a conservative approach with
the aim of reducing misclassification, considering only explicit geographical
information directly provided by the authors. The full set of users geolocated
using the above methodology consists of 1,408,388 users, with state
representativeness in the order of 5.5 Reddit users per thousand U.S.
residents (median value among all U.S. states). Although we acknowledge a
potential bias due to heterogeneities in Reddit population coverage and users
demographics, the number of Reddit users has good linear correlation of $r$ =
0.89 and $p$-value below 1e-12 with census population (Figure 2).

##  6\. Prevalence of interest in opiates

Figure 3. US states distribution maps: choropleth maps representing the
overdose deaths rate for 2016 (red), the opioid prescription rate for 2016
(blue), the opioid interest rate in Reddit for 2016 and 2017 (green).

Conversations in opiates related subreddits branch off in many topics, mostly
regarding opiates usage, dosages, interactions with other substances, safe
practices and withdrawal, usually with a personal perspective. Users share
their health and addiction status and provide support among each other. In
general, they share a common and firsthand interest in opiates experiences.
Thus, the number of authors participating in the conversations in opiates
related subreddits (as identified in Section 4) is not to be considered as a
crude number of opiates users and addicts but it is rather to be considered as
a proxy of users personally interested in opiates in the broadest sense.

Using the geographical information of Reddit population estimated in Section 5
and having identified the opiates authors in Section 4.1, we were able to
evaluate the opiates interest prevalence at the US state level as the fraction
of geolocated users engaged in opiates subreddits and the total estimated
population.

We mapped to US states the 9,026 geolocated users who posted in opioid related
subreddits, equivalent to 24% of the opiates authors, for a mean interest
prevalence of 636 per 100,000 Reddit users (CDC data for 2016 reported an age-
adjusted rate of overdose deaths of 19.8 per 100,000 , and opioid prescription
rate of 66.5 prescriptions dispensed per 100 persons).

|  | Opiates | Reddit | Interest
---|---|---|---|---
|  | authors | authors | prevalence
Region | Division |  |  |
Northeast | Middle Atlantic | 1,186 | 154,418 | 768.05
Northeast | New England | 455 | 69,132 | 658.16
Midwest | East North Central | 1,082 | 172,902 | 625.79
Midwest | West North Central | 424 | 92,931 | 456.25
South | East South Central | 457 | 63,269 | 722.31
South | South Atlantic | 1,656 | 242,470 | 682.97
South | West South Central | 1,079 | 177,856 | 606.67
West | Mountain | 793 | 119,742 | 662.26
West | Pacific | 1,894 | 315,668 | 599.10
Table 1. Interest prevalence by US regional division: number of opiates
authors, total number of authors, and interest prevalence per 100,000
individuals measured on Reddit.

The areas of greater interest according to our estimates is the South Region
(Table 1), with high prevalence for Mississippi, Arkansas, Louisiana and the
highest measured value of 1,180 interested users per 100,000 population in
West Virginia (Figure 3, green map). Middle Atlantic and New England states
like Pennsylvania, New Jersey and Rhode Island are also largely involved,
showing high interest rates ranging between 850 to 900 individuals per
100,000. In line with official statistics about drugs overdose deaths, West
North Central states are those with the lowest interest rate measured on
Reddit, ranging from 341 per 100,000 in Nebraska to 510 per 100,000 in
Minnesota.

We confronted the estimated interest prevalence with official statistics from
the Centers for Disease Control and Prevention
777https://www.cdc.gov/drugoverdose/data/index.html grasping different angles
of the opioid crisis. In particular we focused on opioid drug overdose deaths
rates and retail opioid prescribing rates, both regarding 2016 (the most
recent available dataset at the time of writing), shown in Figure 3 in red and
blue respectively. These two phenomena seem fairly uncorrelated, with a
Pearson’s correlation of 0.068 (p-value of 0.637). It is worth to stress that
those “gold-standard” data are the only ones provided by CDC that allows for
comparisons between different states: although the counting of drugs overdose
deaths includes every drug and is not broken down by drug type, state-level
estimates of opiate-related overdose deaths are affected by heterogeneities in
the surveillance system. On the other hand, official statistics about
prescribing rates that includes both appropriate prescriptions and drug abuse
are affected by different prescription policies in place at different states.

The interest prevalence shows fairly high positive linear correlations with
CDC rates, respectively r = 0.45 (p-value = 8.4e-04) with the opioid overdose
deaths rate and r = 0.506 (p-value = 1.6e-04) with the retail opioid
prescribing rate. These correlations suggest that the signal of interest in
opiates measured on Reddit partially explains the observed phenomena around
opioid epidemics measured by the standard surveillance system. Moreover, we
trained a linear regression model to fit the estimated prevalence using drug
overdose death rates and prescriptions rates as features and predicted new
values of interest prevalence, resulting in an higher correlation of r = 0.655
(p-value = 1.7e-07) with the estimated interest prevalence. This result
confirms that we are sensing a broad signal, tied to drug overdose deaths and
opioid prescriptions but probably accounting for more complex aspects of the
phenomenon.

Figure 4. Opioids interest prevalence: number of Reddit authors per 100,000
Reddit population. Prevalence values for years 2016 and 2017 are reported on
the x-axis. The size of the bubbles is proportional to the number of Reddit
authors.

Leveraging the geolocated cohort, we also evaluated the temporal variation of
interest prevalence between 2016 and 2017 broken down by state. According to
Figure 4, the interest prevalence decreased only in 8 states while in general
we observe that in areas with a good coverage of opiates-related users
(namely, California, Texas, New York, Florida), the interest prevalence
increased by 10% to 20%. It is worth to stress that, at the time of writing,
no official data about 2017 drug overdose deaths and associated trends are
available, highlighting the tremendous potential of a digital epidemiology
approach to gather timely insights about hard-to-reach information of health-
related topics at the population level.

##  7\. Conclusion

This study provides an analysis of Reddit content related to personal opiates
abuse in the period of 2016 and 2017. Starting from almost 2 billion posts
over 74k distinct subreddits, we applied a general information retrieval
algorithm to identify specific subreddits of interest thus selecting 37,009
users that show an explicit interest in the topic. 1.5 million pseudonymous
Reddit users were geolocated at US state level by looking at the content they
generated in the platform. The number of mapped users for each state are in
good agreement with census data, with some differences in terms of coverage.
Such cohort might represent a biased yet valuable digital observatory on
several social, political and health-related topics. The prevalence of opiates
interest extracted with the presented approach shows a complementary
perspective to official surveillance, and its geographical heterogeneity
partially encodes signals from opioid prescribing rates and drug overdose
deaths.

###### Acknowledgements.

DB acknowledges support from the Lagrange Project and CRT Foundation
(http://isi.it/en/lagrange-project/project).

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