This article is accepted for publishing in

the Journal of Fourier and Analysis and Applications

Heyde theorem for locally compact Abelian groups containing no subgroups

topologically isomorphic to the 2-dimensional torus

Gennadiy Feldman (ORCID ID https://orcid.org/0000-0001-5163-4079)

Abstract
We prove the following group analogue of the well-known Heyde theorem on a
characterization of the Gaussian distribution on the real line. Let $X$ be a
second countable locally compact Abelian group containing no subgroups
topologically isomorphic to the 2-dimensional torus. Let $G$ be the subgroup
of $X$ generated by all elements of $X$ of order $2$ and let $\alpha$ be a
topological automorphism of the group $X$ such that ${\rm
Ker}(I+\alpha)=\\{0\\}$. Let $\xi_{1}$ and $\xi_{2}$ be independent random
variables with values in $X$ and distributions $\mu_{1}$ and $\mu_{2}$ with
nonvanishing characteristic functions. If the conditional distribution of the
linear form $L_{2}=\xi_{1}+\alpha\xi_{2}$ given $L_{1}=\xi_{1}+\xi_{2}$ is
symmetric, then $\mu_{j}$ are convolutions of Gaussian distributions on $X$
and distributions supported in $G$. We also prove that this theorem is false
if $X$ is the 2-dimensional torus.

Keywords locally compact Abelian group $\cdot$ 2-dimensional torus $\cdot$
topological automorphism $\cdot$ Gaussian distribution $\cdot$ conditional
distribution

Mathematics Subject Classification Primary: 43A25 $\cdot$ 43A35 $\cdot$ 60B15
$\cdot$ 62E10

##  1 Introduction

The following characterization theorem was proved in [15, Theorem 3.1].

######  Theorem A

Let $X$ be a second countable locally compact Abelian group with the connected
component of zero of dimension $1$. Let $G$ be the subgroup of $X$ generated
by all elements of $X$ of order $2$ and let $\alpha$ be a topological
automorphism of the group $X$ satisfying the condition

${\rm Ker}(I+\alpha)=\\{0\\}.$ (1)

Let $\xi_{1}$ and $\xi_{2}$ be independent random variables with values in $X$
and distributions $\mu_{1}$ and $\mu_{2}$ with nonvanishing characteristic
functions. If the conditional distribution of the linear form
$L_{2}=\xi_{1}+\alpha\xi_{2}$ given $L_{1}=\xi_{1}+\xi_{2}$ is symmetric, then
$\mu_{j}$ are convolutions of Gaussian distributions on $X$ and distributions
supported in $G$.

Theorem A can be considered as a group analogue for two independent random
variables of the well-known theorem of C.C. Heyde ([18], see also [19, §
13.4.1]), where the Gaussian distribution on the real line is characterized by
the symmetry of the conditional distribution of one linear form of independent
random variables given another.

Denote by $\mathbb{R}$ the additive group of real numbers and by $\mathbb{T}$
the circle group (the one-dimensional torus), i.e., the multiplicative group
of all complex numbers with absolute value 1. The aim of the article is,
firstly, to prove that Theorem A is true for a much wider class of locally
compact Abelian groups, namely for second countable locally compact Abelian
groups containing no subgroups topologically isomorphic to the $2$-dimensional
torus $\mathbb{T}^{2}$ (Theorem 2.1). Secondly, to prove that Theorem A is
false when $X$ is the $2$-dimensional torus $\mathbb{T}^{2}$ (Theorem 3.1). We
emphasize that the proof of Theorem 2.1 given in the article is fundamentally
different from the proof of Theorem A, which is based on the proof of the
theorem for the group $X$ of the form $X=\mathbb{R}\times D$, where $D$ is a
second countable totally disconnected locally compact Abelian group, and uses
complex analysis.

Many studies have been devoted to analogues of Heyde’s theorem for different
classes of locally compact Abelian groups (see e.g. [3, 4, 5, 7, 8, 9, 10, 11,
12, 20, 21, 22, 13], and also [14, Chapter IV], where one can find additional
references). In the article we continue this research.

We use in the article standard facts related to abstract harmonic analysis,
see e.g. [17]. Let $X$ be a second countable locally compact Abelian group,
let ${\rm Aut}(X)$ be the group of topological automorphisms of the group $X$,
and let $I$ be the identity automorphism of a group. Denote by $Y=X^{*}$ the
character group of the group $X$, and by $(x,y)$ the value of a character
$y\in Y$ at an element $x\in X$. For a closed subgroup $K$ of the group $X$,
denote by $A(Y,K)=\\{y\in Y:(x,y)=1$ for all $x\in K\\}$ its annihilator. The
character group of the factor-group $X/K$ is topologically isomorphic to the
annihilator $A(Y,K)$. Let $\alpha:X\rightarrow X$ be a continuous endomorphism
of the group $X$. The adjoint endomorphism $\widetilde{\alpha}:Y\rightarrow Y$
is defined as follows: $(\alpha x,y)=(x,\widetilde{\alpha}y)$ for all $x\in
X$, $y\in Y$. Let $n$ be a natural number. Put $X^{(n)}=\\{nx:x\in X\\}$. A
topological isomorphism of locally compact Abelian groups $X_{1}$ and $X_{2}$
is denoted as $X_{1}\cong X_{2}$. Denote by $\mathbb{Z}$ the additive group of
integers.

Let $f(y)$ be a function on the group $Y$ and let $h\in Y$. Denote by
$\Delta_{h}$ the finite difference operator

$\Delta_{h}f(y)=f(y+h)-f(y),\quad y\in Y.$

Denote by ${\rm M}^{1}(X)$ the convolution semigroup of probability
distributions on the group $X$. Let $\mu\in{\rm M}^{1}(X)$. Denote by

$\hat{\mu}(y)=\int_{X}(x,y)d\mu(x),\quad y\in Y,$

the characteristic function (Fourier transform) of the distribution $\mu$. The
characteristic function of a signed measure is defined in the same way. Denote
by $\sigma(\mu)$ the support of $\mu$. Define the distribution
$\bar{\mu}\in{\rm M}^{1}(X)$ by the formula $\bar{\mu}(B)=\mu(-B)$ for any
Borel subset $B$ in $X$. Then $\hat{\bar{\mu}}(y)=\overline{\hat{\mu}(y)}$. If
$F$ is a Borel subgroup of $X$, denote by ${\rm M}^{1}(F)$ the subsemigroup of
${\rm M}^{1}(X)$ consisting of all distributions concentrated on $F$.

A distribution $\gamma\in{\rm M^{1}}(X)$ is called Gaussian ([24, Chapter
IV, §6]) if its characteristic function can be represented in the form

$\hat{\gamma}(y)=(x,y)\exp\\{-\varphi(y)\\},\quad y\in Y,$ (2)

where $x\in X$ and $\varphi(y)$ is a continuous nonnegative function on the
group $Y$ satisfying the equation

$\varphi(u+v)+\varphi(u-v)=2[\varphi(u)+\varphi(v)],\quad u,v\in Y.$ (3)

A Gaussian distribution is called symmetric if $x=0$ in (2). Denote by
$\Gamma(X)$ the set of Gaussian distributions on the group $X.$ Note that in
particular, the degenerate distributions are Gaussian. Let $x\in X$. Denote by
$E_{x}$ the degenerate distribution concentrated at the point $x\in X$.

Denote by $m_{X}$ a Haar measure on the group $X$. It is well known that
$m_{X}$ is unique up to a positive multiplicative constant. If $X$ is a
compact group, then $m_{X}(X)<\infty$. We suppose that in this case
$m_{X}\in{\rm M^{1}}(X)$. Note that, if $X$ is an arbitrary locally compact
Abelian group and $K$ is a compact subgroup of $X$, then the characteristic
function $\widehat{m}_{K}(y)$ is of the form

$\widehat{m}_{K}(y)=\begin{cases}1,&\text{\ if\ }\ \ y\in A(Y,K),\\\
0,&\text{\ if\ }\ \ y\not\in A(Y,K).\end{cases}$ (4)

##  2 Proof of the main theorem

The main result of the article is the following theorem.

######  Theorem 2.1

Let $X$ be a second countable locally compact Abelian group containing no
subgroups topologically isomorphic to the $2$-dimensional torus
$\mathbb{T}^{2}$. Let $G$ be the subgroup of $X$ generated by all elements of
$X$ of order $2$ and let $\alpha$ be a topological automorphism of the group
$X$ satisfying condition $(\ref{d1})$. Let $\xi_{1}$ and $\xi_{2}$ be
independent random variables with values in the group $X$ and distributions
$\mu_{1}$ and $\mu_{2}$ with nonvanishing characteristic functions. If the
conditional distribution of the linear form $L_{2}=\xi_{1}+\alpha\xi_{2}$
given $L_{1}=\xi_{1}+\xi_{2}$ is symmetric, then $\mu_{j}\in\Gamma(X)*{\rm
M^{1}}(G)$, $j=1,2$.

To prove Theorem 2.1 we need some lemmas.

######  Lemma 2.2

([21, Lemma 3.8], see also [14, Corollary 9.7]) Let $X$ be a second
countable locally compact Abelian group and let $\alpha$ be a topological
automorphism of $X$. Let $\xi_{1}$ and $\xi_{2}$ be independent random
variables with values in the group $X$. If the conditional distribution of the
linear form $L_{2}=\xi_{1}+\alpha\xi_{2}$ given $L_{1}=\xi_{1}+\xi_{2}$ is
symmetric, then the linear forms $P_{1}=(I+\alpha)\xi_{1}+2\alpha\xi_{2}$ and
$P_{2}=2\xi_{1}+(I+\alpha)\xi_{2}$ are independent.

######  Lemma 2.3

([6, Lemma 10.1]) Let $X$ be a second countable locally compact Abelian
group with character group $Y$. Let $\alpha_{j}$, $\beta_{j}$, $j=1,2$, be
continuous endomorphisms of the group $X$. Let $\xi_{1}$ and $\xi_{2}$ be
independent random variables with values in the group $X$ and distributions
$\mu_{1}$ and $\mu_{2}$. The linear forms
$L_{1}=\alpha_{1}\xi_{1}+\alpha_{2}\xi_{2}$ and
$L_{2}=\beta_{1}\xi_{1}+\beta_{2}\xi_{2}$ are independent if and only if the
characteristic functions $\hat{\mu}_{j}(y)$ satisfy the equation

$\hat{\mu}_{1}(\widetilde{\alpha}_{1}u+\widetilde{\beta}_{1}v)\hat{\mu}_{2}(\widetilde{\alpha}_{2}u+\widetilde{\beta}_{2}v)=\hat{\mu}_{1}(\widetilde{\alpha}_{1}u)\hat{\mu}_{2}(\widetilde{\alpha}_{2}u)\hat{\mu}_{1}(\widetilde{\beta}_{1}v)\hat{\mu}_{2}(\widetilde{\beta}_{2}v),\quad
u,v\in Y.$ (5)

######  Lemma 2.4

([7, Lemma 6], see also [14, Lemma 5.1]) Let $Y$ be a locally compact
Abelian group and let $A(y)$ be a continuous function on the group $Y$
satisfying the equation

$\Delta_{2k}\Delta^{2}_{h}A(y)=0,\quad y,k,h\in Y,$

and the conditions $A(-y)=A(y)$, $A(0)=0$. Let

$Y=\bigcup_{\iota}{(y_{\iota}+\overline{Y^{(2)}})},\quad y_{0}=0,$ (6)

be a $\overline{Y^{(2)}}$-coset decomposition of the group $Y$. Then the
function $A(y)$ can be represented in the form

$A(y)=\varphi(y)+r_{\iota},\quad y\in y_{\iota}+\overline{Y^{(2)}},$ (7)

where $\varphi(y)$ is a continuous function on the group $Y$ satisfying
equation $(\ref{09.07.1})$.

Let us recall some definitions. Let $X$ be an Abelian group. We do not assume
that $X$ is a topological group. If each element of $X$ has finite order, then
we say that $X$ is a torsion group. We say that $X$ is a torsion-free group if
each element of $X$, except zero, has infinite order. The subgroup consisting
of all elements of finite order of the group $X$ is called a torsion part of
$X$.

We formulate as a lemma the following corollary of the well-known Baer–Fomin
theorem ([16, Theorem 100.1]).

######  Lemma 2.5

Let $Y$ be an Abelian group and let $H$ be a torsion part of $Y$. If $H$ is a
bounded subgroup, then $H$ is a direct factor of $Y$.

Using Lemma 2.5 we shall prove the following statement.

######  Lemma 2.6

Let $X$ be a locally compact Abelian group containing no subgroups
topologically isomorphic to the circle group $\mathbb{T}$. Let $K$ be a closed
bounded subgroup of $X$. Then the factor-group $X/K$ also contains no
subgroups topologically isomorphic to the circle group $\mathbb{T}$.

###### Proof.

The lemma is true if $K=X$, so we suppose that $K\neq X$. Assume that the
factor-group $X/K$ contains a subgroup $F$ such that $F\cong\mathbb{T}$. Let
us prove that in this case we arrive at a contradiction. Let $p:X\rightarrow
X/K$ be the natural homomorphism. Put $S=p^{-1}(F)$. The group $S$ is closed
and hence locally compact. Consider the restriction of $p$ to the subgroup
$S$. We have $p:S\rightarrow S/K$ and $S/K\cong\mathbb{T}$. Thus, we can prove
the lemma assuming that $X/K\cong\mathbb{T}$. By the structure theorem for
locally compact Abelian groups, the group $X$ is topologically isomorphic to a
group of the form $\mathbb{R}^{m}\times G$, where a locally compact Abelian
group $G$ contains a compact open subgroup. Since $K$ is a bounded subgroup,
$K\subset G$. It follows from $X/K\cong\mathbb{T}$ that $m=0$, i.e., the group
$X$ itself contains a compact open subgroup. Denote by $Y$ the character group
of the group $X$.

First suppose that $X$ is a compact group. Then $Y$ is a discrete group.
Denote by $H$ the torsion part of the group $Y$. We have
$A(Y,K)\cong(X/K)^{*}$. Since $X/K\cong\mathbb{T}$, this implies that

$A(Y,K)\cong\mathbb{Z}.$ (8)

Inasmuch as $K$ is a bounded group, there is a natural $n$ such that $nx=0$
for all $x\in K$. It follows from this that $ny\in A(Y,K)$ for each $y\in Y$,
i.e.,

$Y^{(n)}\subset A(Y,K).$ (9)

In particular,

$H^{(n)}\subset A(Y,K).$ (10)

In view of (8), (10) implies that

$H^{(n)}=\\{0\\}.$ (11)

Hence $H$ is a bounded group. By Lemma 2.5, $H$ is a direct factor of $Y$. We
have $Y=H\times L$, where $L$ is a torsion-free group. Taking into account
(11), this implies that $Y^{(n)}=L^{(n)}$. For this reason, it follows from
(8) and (9) that $L\cong\mathbb{Z}$. This implies that the group $X$ contains
a subgroup topologically isomorphic to the circle group $\mathbb{T}$, contrary
to the assumption. Thus, the lemma is proved if the group $X$ is compact.

Consider the general case. Let $B$ be a compact open subgroup of the group
$X$. As far as $p$ is a continuous open epimorphism, $p(B)$ is an open
subgroup of the factor-group $X/K$. Since $X/K\cong\mathbb{T},$ we have
$p(B)=X/K.$ Moreover, $p(B)\cong B/(K\cap B)$. Thus, $B/(K\cap
B)\cong\mathbb{T}$. As has been shown above, this implies that the group $B$,
and hence the group $X$, contains a subgroup topologically isomorphic to the
circle group $\mathbb{T}$, contrary to the assumption. ∎

Let $\mathbb{R}^{\aleph_{0}}$ be the space of all sequences of real numbers
considering in the product topology. We will need the definition of the
Gaussian distribution in $\mathbb{R}^{\aleph_{0}}$. The space
$\mathbb{R}^{\aleph_{0}}$ is one of the simplest examples of a locally convex
space. We note that Gaussian distributions in arbitrary locally convex spaces
are studied in details in the fundamental monograph [1].

Denote by $\mathbb{R}^{\aleph_{0}*}$ the space of all finitary sequences of
real numbers with the topology of strictly inductive limit of spaces
$\mathbb{R}^{n}$. Let
${t}=(t_{1},t_{2},\dots,t_{n},\dots)\in\mathbb{R}^{\aleph_{0}}$ and
${s}=(s_{1},s_{2},\dots,s_{n},0,\dots)\in\mathbb{R}^{\aleph_{0}*}$. Set

$\langle{t},{s}\rangle=\mathop{\sum}\limits_{j=1}^{\infty}t_{j}s_{j}.$

Let $\mu$ be a distribution on $\mathbb{R}^{\aleph_{0}}$. We define the
characteristic function of $\mu$ by the formula

$\hat{\mu}({s})=\int\limits_{\mathbb{R}^{\aleph_{0}}}\exp\\{i\langle{t},{s}\rangle\\}d\mu({t}),\quad{s}\in\mathbb{R}^{\aleph_{0}*}.$

Let $A=(\alpha_{ij})_{i,j=1}^{\infty}$ be a symmetric positive semidefinite
matrix, i.e., the quadratic form

$\langle
As,{s}\rangle=\mathop{\sum}\limits_{i,j=1}^{\infty}\alpha_{ij}s_{i}s_{j}$

is nonnegative for all
${s}=(s_{1},s_{2},\dots,s_{n},0,\dots)\in\mathbb{R}^{\aleph_{0}*}$.

A distribution $\mu$ on the group $\mathbb{R}^{\aleph_{0}}$ is called Gaussian
if its characteristic function is represented in the form

$\hat{\mu}({s})=\exp\\{i\langle{t},{s}\rangle-\langle
A{s},{s}\rangle\\},\quad{s}\in\mathbb{R}^{\aleph_{0}*},$

where ${t}\in\mathbb{R}^{\aleph_{0}}$ and $A=(\alpha_{ij})_{i,j=1}^{\infty}$
is a symmetric positive semidefinite matrix.

######  Lemma 2.7

([2], see also [6, §3]) Let $X$ be a second countable locally compact
Abelian group containing no subgroups topologically isomorphic to the circle
group $\mathbb{T}.$ Then there exists a continuous monomorphism
$p:E\rightarrow X$, where either $E=\mathbb{R}^{n}$ for some $n$ or
$E=\mathbb{R}^{\aleph_{0}}$, such that if $\gamma$ is a symmetric Gaussian
distribution on $X$, then $\gamma=p(M)$, where $M$ is a symmetric Gaussian
distribution on $E$.

######  Lemma 2.8

Let $X$ be a second countable locally compact Abelian group containing no
subgroups topologically isomorphic to the circle group $\mathbb{T}.$ Let $G$
be the subgroup of $X$ generated by all elements of $X$ of order $2$. Let
$\mu\in\Gamma(X)*{\rm M}^{1}(G)$ and suppose that the characteristic function
of the distribution $\mu$ does not vanish. If $\mu=\mu_{1}*\mu_{2}$, where
$\mu_{j}\in{\rm M}^{1}(X)$, then $\mu_{j}\in\Gamma(X)*{\rm M}^{1}(G)$,
$j=1,2$.

###### Proof.

Since the group $X$ contains no subgroups topologically isomorphic to the
circle group $\mathbb{T}$, let $p$ be a continuous monomorphism
$p:E\rightarrow X$ which exists by Lemma 2.7. As far as $p$ is a monomorphism,
we have $p(E)\cap G=\\{0\\}$. Hence $p$ can be extended to a continuous
monomorphism $\bar{p}:E\times G\rightarrow X$ by the formula
$\bar{p}(t,g)=p(t)+g$, $t\in E$, $g\in G$, and $\bar{p}$ generates an
isomorphism of the semigroups ${\rm M}^{1}(E\times G)$ and ${\rm
M}^{1}(p(E)\times G)$. Let $\mu=\gamma*\omega$, where $\gamma\in\Gamma(X)$,
$\omega\in{\rm M}^{1}(G)$. We can assume without loss of generality that
$\gamma$ is a symmetric Gaussian distribution. By Lemma 2.7, $\gamma=p(M)$,
where $M\in\Gamma(E)$. Hence $\mu=\bar{p}(N)$, where $N=M*\omega$. Obviously,
the distribution $\mu$ is concentrated on the Borel subgroup $p(E)\times G$ of
the group $X$. Substituting, if it is necessary, the distributions $\mu_{j}$
by their shifts, we can assume that $\mu_{j}$ are also concentrated on the
Borel subgroup $p(E)\times G$. Since the semigroups ${\rm M}^{1}(E\times G)$
and ${\rm M}^{1}(p(E)\times G)$ are isomorphic, we have
$\mu_{j}=\bar{p}(N_{j})$, where $N_{j}\in{\rm M^{1}}(E\times G)$ and $N_{j}$
is a factor of $N$. By [4, Lemma 5 and Remark 4], we have
$N_{j}=M_{j}*\omega_{j}$, where $M_{j}\in\Gamma(E)$, $\omega_{j}\in{\rm
M^{1}}(G)$. Hence

$\mu_{j}=\bar{p}(N_{j})=\bar{p}(M_{j}*\omega_{j})=\bar{p}(M_{j})*\bar{p}(\omega_{j})=p(M_{j})*\omega_{j}=\gamma_{j}*\omega_{j},$

where $\gamma_{j}=p(M_{j})$. Since $\gamma_{j}\in\Gamma(X)$, the lemma is
proved. ∎

Proof of Theorem 2.1 Assume that a locally compact Abelian group $X$ contains
no subgroups topologically isomorphic to the 2-dimensional torus
$\mathbb{T}^{2}$ but contains a subgroup topologically isomorphic to the
circle group $\mathbb{T}$. Obviously, there can be only one such subgroup.
Hence it is invariant with respect to each topological automorphism of the
group $X$. Since ${\rm Aut}(\mathbb{T})=\\{\pm I\\}$, this implies that there
is no topological automorphism $\alpha$ of the group $X$ satisfying condition
$(\ref{d1})$. Hence we can prove the theorem assuming from the beginning that
the group $X$ contains no subgroups topologically isomorphic to the circle
group $\mathbb{T}$.

Denote by $Y$ the character group of the group $X$. By Lemma 2.2, the symmetry
of the conditional distribution of the linear form $L_{2}$ given $L_{1}$
implies that the linear forms $P_{1}=(I+\alpha)\xi_{1}+2\alpha\xi_{2}$ and
$P_{2}=2\xi_{1}+(I+\alpha)\xi_{2}$ are independent. By Lemma 2.3, it follows
from this that the characteristic functions $\hat{\mu}_{j}(y)$ satisfy
equation $(\ref{4})$, which takes the form

$\hat{\mu}_{1}((I+\widetilde{\alpha})u+2v)\hat{\mu}_{2}(2\widetilde{\alpha}u+(I+\widetilde{\alpha})v)\\\
=\hat{\mu}_{1}((I+\widetilde{\alpha})u)\hat{\mu}_{2}(2\widetilde{\alpha}u)\hat{\mu}_{1}(2v)\hat{\mu}_{2}((I+\widetilde{\alpha})v),\quad
u,v\in Y.$ (12)

Put $\nu_{j}=\mu_{j}*\bar{\mu}_{j}$. We have
$\hat{\nu}_{j}(y)=|\hat{\mu}_{j}(y)|^{2}>0$ for all $y\in Y$, $j=1,2$. The
characteristic functions $\hat{\nu}_{j}(y)$ also satisfy equation $(\ref{5})$.
Set $\psi_{j}(y)=-\ln\hat{\nu}_{j}(y)$, $j=1,2$. It follows from $(\ref{5})$
that the functions $\psi_{j}(y)$ satisfy the equation

$\psi_{1}((I+\widetilde{\alpha})u+2v)+\psi_{2}(2\widetilde{\alpha}u+(I+\widetilde{\alpha})v)=A(u)+B(v),\quad
u,v\in Y,$ (13)

where

$A(y)=\psi_{1}((I+\widetilde{\alpha})y)+\psi_{2}(2\widetilde{\alpha}y),\quad
B(y)=\psi_{1}(2y)+\psi_{2}((I+\widetilde{\alpha})y),\quad y\in Y.$ (14)

Equation (13) has already appeared earlier in the study of Heyde’s theorem on
various locally compact Abelian groups (see e.g. [10], [11]). For
completeness, we give here its solution. We use the finite difference method.
Take an arbitrary element $h_{1}$ of the group $Y$. Substitute
$u+(I+\widetilde{\alpha})h_{1}$ for $u$ and $v-2\widetilde{\alpha}h_{1}$ for
$v$ in equation (13). Subtracting equation (13) from the obtaining equation we
get

$\Delta_{(I-\widetilde{\alpha})^{2}h_{1}}{\psi_{1}((I+\widetilde{\alpha})u+2v)}=\Delta_{(I+\widetilde{\alpha})h_{1}}A(u)+\Delta_{-2\widetilde{\alpha}h_{1}}B(v),\quad
u,v\in Y.$ (15)

Take an arbitrary element $h_{2}$ of the group $Y$. Substitute $u+2h_{2}$ for
$u$ and $v-(I+\widetilde{\alpha})h_{2}$ for $v$ in equation (15). Subtracting
equation (15) from the resulting equation we obtain

$\Delta_{2h_{2}}\Delta_{(I+\widetilde{\alpha})h_{1}}A(u)+\Delta_{-(I+\widetilde{\alpha})h_{2}}\Delta_{-2\widetilde{\alpha}h_{1}}B(v)=0,\quad
u,v\in Y.$ (16)

Take an arbitrary element $h$ of the group $Y$. Substituting $u+h$ for $u$ in
equation (16) and subtracting equation (16) from the obtaining equation we get

$\Delta_{h}\Delta_{2h_{2}}\Delta_{(I+\widetilde{\alpha})h_{1}}A(u)=0,\quad
u\in Y.$ (17)

It follows from properties of adjoint homomorphisms that if a topological
automorphism $\alpha$ satisfies condition $(\ref{d1})$, then the subgroup
$(I+\widetilde{\alpha})(Y)$ is dense in $Y$. Taking into account that
$h,h_{1},h_{2}$ are arbitrary elements of the group $Y$, it follows from (17)
that the function $A(y)$ satisfies the equation

$\Delta_{2k}\Delta_{h}^{2}A(y)=0,\quad y,k,h\in Y.$ (18)

By Lemma 2.4, (18) implies that the function $A(y)$ is represented in the form
(7), where the function $\varphi(y)$, as is easily seen, is nonnegative.
Denote by $\mu$ the distribution of the random variable
$P_{1}=(I+\alpha)\xi_{1}+2\alpha\xi_{2}$. It is obvious that

$\mu=(I+\alpha)(\mu_{1})*(2\alpha)(\mu_{2}).$

Put $\nu=\mu*\bar{\mu}$. Then

$\nu=(I+\alpha)(\mu_{1})*(2\alpha)(\mu_{2})*(I+\alpha)(\bar{\mu}_{1})*(2\alpha)(\bar{\mu}_{2}).$
(19)

It follows from (14) and (19) that the characteristic function $\hat{\nu}(y)$
is of the form

$\hat{\nu}(y)=e^{-A(y)},\quad y\in Y.$ (20)

Denote by $\gamma$ the Gaussian distribution on the group $X$ with the
characteristic function

$\hat{\gamma}(y)=\exp\\{-\varphi(y)\\},\quad y\in Y.$ (21)

Taking into account (6) and (7), define on the group $Y$ the function $g(y)$
by the formula

$g(y)=\exp\\{-r_{\iota}\\},\quad y\in y_{\iota}+\overline{Y^{(2)}}.$ (22)

Since $g(y)=\hat{\nu}(y)/\hat{\gamma}(y)$, the function $g(y)$ is continuous.
Moreover, (22) implies that the function $g(y)$ is invariant with respect to
the subgroup $\overline{Y^{(2)}}$. Check that $g(y)$ is a positive definite
function. Note that

$A(X,\overline{Y^{(2)}})=G.$ (23)

Consider decomposition (6) and take a finite set of elements $y_{\iota_{j}}$,
$j=1,2,\dots,n$. Let $H$ be a subgroup of $Y$ generated all cosets
$y_{\iota_{j}}+\overline{Y^{(2)}}$. Obviously, it suffices to verify that the
restriction of the function $g(y)$ to $H$ is a positive definite function. The
subgroup $H$ consists of a finite number of cosets
$y_{\iota}+\overline{Y^{(2)}}$. Put $K=A(X,H)$. Then $H^{*}\cong X/K$ and
obviously, $K\subset G$. Consider the restriction of the function $g(y)$ to
$H$. This restriction is invariant with respect to the subgroup
$\overline{Y^{(2)}}$ and hence defines a function on the factor-group
$H/\overline{Y^{(2)}}$. We note that $H/\overline{Y^{(2)}}$ is a finite group
and all its nonzero elements are of the order 2. It follows from this that any
real-valued function on the factor-group $H/\overline{Y^{(2)}}$ is the
characteristic function of a signed measure. In particular, the restriction of
the function $g(y)$ to the subgroup $H$ is the characteristic function of a
signed measure $\varpi$. Since $(H/\overline{Y^{(2)}})^{*}\cong
A(X/K,\overline{Y^{(2)}})$, we can consider the signed measure $\varpi$ as a
signed measure on the finite subgroup $F=A(X/K,\overline{Y^{(2)}})$. It
follows from (7) and (20)–(22) that the restriction of the characteristic
function $\hat{\nu}(y)$ to $H$ is the characteristic function of the
convolution of a Gaussian distribution $\lambda$ on $X/K$ and the signed
measure $\varpi$ on $F$. We verify that the signed measure $\varpi$ is
actually a distribution and this proves that $g(y)$ is a positive definite
function.

Since $K\subset G$, by Lemma 2.6, the factor-group $X/K$ contains no subgroups
topologically isomorphic to the circle group $\mathbb{T}$. Then by Lemma 2.7,
applying to the group $X/K$, there exists a continuous monomorphism
$p:E\rightarrow X/K$, where either $E=\mathbb{R}^{n}$ for some $n$ or
$E=\mathbb{R}^{\aleph_{0}}$, such that $\lambda=p(M)$, where $M$ is a
symmetric Gaussian distribution on $E$. Hence the Gaussian distribution
$\lambda$ is concentrated on the Borel subgroup $p(E)$ of the group $X/K$.
Note that all nonzero elements of the group $F$ are of the order 2 and hence

$p(E)\cap F=\\{0\\}.$ (24)

Since the convolution $\lambda*\varpi$ is a distribution, in view of (24),
$\varpi$ is also a distribution. Thus, we proved that $g(y)$ is a continuous
positive definite function such that $g(0)=0$. By the Bochner theorem, there
exists a distribution $\omega\in{\rm M}^{1}(X)$ such that
$\hat{\omega}(y)=g(y)$, $y\in Y$. Since $g(y)=1$ for $y\in\overline{Y^{(2)}}$,
in view of (23), we have an inclusion

$\sigma(\omega)\subset A\left(X,\\{y\in Y:g(y)=1\\}\right)\subset
A(X,\overline{Y^{(2)}})=G.$

It follows from $\hat{\nu}(y)=\hat{\gamma}(y)g(y)$ for all $y\in Y$, that
$\nu=\gamma*\omega\in\Gamma(X)*{\rm M}^{1}(G)$. By Lemma 2.8, (19) implies
that $(I+\alpha)(\mu_{1})\in\Gamma(X)*{\rm M}^{1}(G)$. Taking into account
that $I+\alpha$ is a continuous monomorphism, we get $\mu_{1}\in\Gamma(X)*{\rm
M}^{1}(G)$.

To complete the proof of the theorem, it remains to prove that
$\mu_{2}\in\Gamma(X)*{\rm M}^{1}(G)$. Take an arbitrary element $k$ of the
group $Y$ and substitute $v+k$ for $u$ in (16). Subtracting equation (16) from
the resulting equation, we obtain

$\Delta_{k}\Delta_{-(I+\widetilde{\alpha})h_{2}}\Delta_{-2\widetilde{\alpha}h_{1}}B(v)=0,\quad
v\in Y.$

Consider the distribution of the random variable
$P_{2}=2\xi_{1}+(I+\alpha)\xi_{2}$. Arguing in the same way as in the case
when we considered the distribution of the random variable $P_{1}$, we prove
that $\mu_{2}\in\Gamma(X)*{\rm M}^{1}(G)$.

$\Box$

Evidently, if the connected component of zero of a second countable locally
compact Abelian group $X$ has dimension $1$, then $X$ contains no subgroups
topologically isomorphic to the 2-dimensional torus $\mathbb{T}^{2}$. For this
reason Theorem A follows from Theorem 2.1.

We note that the statement of Theorem 2.1 in the case if a topological
automorphism $\alpha$ of the group $X$ satisfies the conditions

$I\pm\alpha\in{\rm Aut(X)}$ (25)

follows from [7, Theorem 1].

It is obvious that if a locally compact Abelian group $X$ contains no elements
of order 2, then $X$ contains no subgroups topologically isomorphic to the
2-dimensional torus $\mathbb{T}^{2}$. Hence Theorem 2.1 implies the following
characterization of the Gaussian distribution proved earlier in [10, Theorem
3], see also [14, Theorem 9.9].

######  Corollary 2.9

Let $X$ be a second countable locally compact Abelian group containing no
elements of order $2$. Let $\alpha$ be a topological automorphism of the group
$X$ satisfying condition $(\ref{d1})$. Let $\xi_{1}$ and $\xi_{2}$ be
independent random variables with values in the group $X$ and distributions
$\mu_{1}$ and $\mu_{2}$ with nonvanishing characteristic functions. Then the
symmetry of the conditional distribution of the linear form
$L_{2}=\xi_{1}+\alpha\xi_{2}$ given $L_{1}=\xi_{1}+\xi_{2}$ implies that
$\mu_{j}\in\Gamma(X)$, $j=1,2$.

Let $X$ be a second countable locally compact Abelian group, let $G$ be the
subgroup of $X$ generated by all elements of $X$ of order 2, and let $\alpha$
be a topological automorphism of the group $X$. Suppose that $K={\rm
Ker}(I+\alpha)\neq\\{0\\}$. Let $\xi_{1}$ and $\xi_{2}$ be independent
identically distributed random variables with values in the subgroup $K$ and
distribution $\mu$. It is obvious that $\alpha x=-x$ for all $x\in K$. It is
easy to see that the conditional distribution of the linear form
$P_{2}=\xi_{1}-\xi_{2}$ given $P_{1}=\xi_{1}+\xi_{2}$ is symmetric. Hence if
we consider independent random variables $\xi_{1}$ and $\xi_{2}$ as
independent random variables taking values in $X$, then the conditional
distribution of the linear form $L_{2}=\xi_{1}+\alpha\xi_{2}$ given
$L_{1}=\xi_{1}+\xi_{2}$ is also symmetric. Since $\mu$ is an arbitrary
distribution, from what has been said it follows that a necessary condition
for a topological automorphism $\alpha$ for Theorem 2.1 to be true is an
inclusion

${\rm Ker}(I+\alpha)\subset G.$ (26)

However, generally speaking, this condition is not sufficient. The
corresponding example can be constructed in the case, when the group $X$ is an
${\bm{a}}$-adic solenoid.

Recall the definition of an ${\bm{a}}$-adic solenoid. Let
${\bm{a}}=(a_{0},a_{1},\dots,a_{n},\dots,)$, where all $a_{j}\in{\mathbb{Z}}$
and $a_{j}>1$. Let $\Delta_{\bm{a}}$ be the group of ${\bm{a}}$-adic integers
and let $B$ be the subgroup of the group $\mathbb{R}\times\Delta_{\bm{a}}$ of
the form $B=\\{(n,n{u})\\}_{n=-\infty}^{\infty}$, where
${u}=(1,0,\dots,0,\dots)$. The factor-group
$\Sigma_{\bm{a}}=(\mathbb{R}\times\Delta_{\bm{a}})/B$ is called an
${\bm{a}}$-adic solenoid (see e.g. [17, §10]). The group $\Sigma_{\bm{a}}$ is
compact, connected, has dimension 1, and contains no subgroups topologically
isomorphic to the circle group $\mathbb{T}$. Moreover, $\Sigma_{\bm{a}}$ can
contain at most one element of order 2. Assume that the group
$\Sigma_{\bm{a}}$ contains an element of order 2 and denote by $G$ the
subgroup of $\Sigma_{\text{\boldmath$a$}}$ generated by this element. It
follows from the results proved in [12], see also [14, Proposition 11.19,
Theorem 11.20 and Remark 11.22]) that the following statement holds.

######  Proposition 2.10

Let $\Sigma_{\bm{a}}$ be an ${\bm{a}}$-adic solenoid containing an element of
order $2$. Let $\alpha$ be a topological automorphism of the group
$\Sigma_{\bm{a}}$ such that

${\rm Ker}(I+\alpha)=G$ (27)

and hence condition $(\ref{9a})$ holds. Then there exist independent random
variables $\xi_{1}$ and $\xi_{2}$ with values in the group $\Sigma_{\bm{a}}$
and distributions $\mu_{1}$ and $\mu_{2}$ with nonvanishing characteristic
functions such that the conditional distribution of the linear form
$L_{2}=\xi_{1}+\alpha\xi_{2}$ given $L_{1}=\xi_{1}+\xi_{2}$ is symmetric,
whereas $\mu_{j}\notin\Gamma(\Sigma_{\bm{a}})*{\rm M^{1}}(G)$, $j=1,2$.

An example of an ${\bm{a}}$-adic solenoid $\Sigma_{\bm{a}}$ and a topological
automorphism $\alpha\in{\rm Aut}(\Sigma_{\bm{a}})$ such that (27) holds is
$\Sigma_{\bm{a}}$, where ${\bm{a}}=(3,3,\dots,3,\dots)$, and $\alpha$ is the
multiplication by $-3$.

##  3 Theorem A is false for the 2-dimensional torus $\mathbb{T}^{2}$

We prove in this section that Theorem A is false for the 2-dimensional torus
$\mathbb{T}^{2}$. Let $X=\mathbb{T}^{2}$. Denote by $x=(z,w)$,
$z,w\in\mathbb{T}$, elements of the group $X$. The character group $Y$ of the
group $X$ is topologically isomorphic to the group $\mathbb{Z}^{2}$. Denote by
$y=(m,n)$, $m,n\in\mathbb{Z}$, elements of the group $Y$. Every automorphism
$\alpha\in{\rm Aut}(X)$ is defined by an integer-valued matrix
$\left(\begin{matrix}a&b\\\ c&d\end{matrix}\right)$, where $|ad-bc|=1$ and
$\alpha$ acts on $X$ as follows

$\alpha(z,w)=(z^{a}w^{c},z^{b}w^{d}),\quad(z,w)\in X.$

The adjoint automorphism $\widetilde{\alpha}\in{\rm Aut}(Y)$ is of the form

$\widetilde{\alpha}(m,n)=(am+bn,cm+dn),\quad(m,n)\in Y.$

We identify $\alpha$ with the matrix $\left(\begin{matrix}a&b\\\
c&d\end{matrix}\right)$ and $\widetilde{\alpha}$ with the matrix
$\left(\begin{matrix}a&c\\\ b&d\end{matrix}\right)$.

It follows from the definition of the Gaussian distribution on a locally
compact Abelian group that the characteristic function of a symmetric Gaussian
distribution on the group $X$ is of the form

$\hat{\gamma}(m,n)=\exp\\{-\langle A(m,n),(m,n)\rangle\\},\quad(m,n)\in Y,$

where $\langle\cdot,\cdot\rangle$ is the standard scalar product in
$\mathbb{R}^{2}$, $A=(a_{ij})_{i,j=1}^{2}$ is a symmetric positive
semidefinite matrix.

######  Theorem 3.1

Let $X=\mathbb{T}^{2}$ and let $G$ be the subgroup of $X$ generated by all
elements of $X$ of order $2$. Then there exist a topological automorphism
$\alpha$ of the group $X$ satisfying condition $(\ref{d1})$ and independent
random variables $\xi_{1}$ and $\xi_{2}$ with values in $X$ and distributions
$\mu_{1}$ and $\mu_{2}$ with nonvanishing characteristic functions such that
the conditional distribution of the linear form $L_{2}=\xi_{1}+\alpha\xi_{2}$
given $L_{1}=\xi_{1}+\xi_{2}$ is symmetric, whereas
$\mu_{j}\notin\Gamma(X)*{\rm M^{1}}(G)$, $j=1,2$.

For the proof of Theorem 3.1 we need the following lemmas.

######  Lemma 3.2

([23], see also [6, Lemma 11.2]) Consider the $2$-dimensional torus
$\mathbb{T}^{2}$. Let $\alpha=\left(\begin{matrix}a&b\\\
c&d\end{matrix}\right)$ be a topological automorphism of the group
$\mathbb{T}^{2}$. If $ad-bc=1$ and $a+d<-2$, then there exist symmetric
positive semidefinite $2\times 2$ matrices $A_{1}$ and $A_{2}$ such that

$\det A_{1}=\det A_{2}>0$ (28)

and

$A_{1}+A_{2}\widetilde{\alpha}=0.$ (29)

######  Lemma 3.3

([6, Lemma 16.1]) Let $X$ be a second countable locally compact Abelian
group with character group $Y$. Let $\alpha$ be a topological automorphism of
$X$. Let $\xi_{1}$ and $\xi_{2}$ be independent random variables with values
in the group $X$ and distributions $\mu_{1}$ and $\mu_{2}$. The conditional
distribution of the linear form $L_{2}=\xi_{1}+\alpha\xi_{2}$ given
$L_{1}=\xi_{1}+\xi_{2}$ is symmetric if and only if the characteristic
functions $\hat{\mu}_{j}(y)$ satisfy the equation

$\hat{\mu}_{1}(u+v)\hat{\mu}_{2}(u+\widetilde{\alpha}v)=\hat{\mu}_{1}(u-v)\hat{\mu}_{2}(u-\widetilde{\alpha}v),\
\ u,v\in Y.$ (30)

Proof of Theorem 3.1 Consider a topological automorphism
$\alpha=\left(\begin{matrix}a&b\\\ c&d\end{matrix}\right)$ of the group $X$
such that $\det\alpha=1$ and $a+d=-3$. It follows from this that
$\det(I+\alpha)=-1$ and hence $I+\alpha\in{\rm Aut}(X)$. For this reason
condition (1) holds. By Lemma 3.2, there exist symmetric positive semidefinite
$2\times 2$ matrices $A_{1}$ and $A_{2}$ such that (28) and (29) are valid.

Note that $\det(I-\alpha)=\det(I-\widetilde{\alpha})=5$. Hence
$I-\widetilde{\alpha}\notin{\rm Aut}(X)$ and it follows from this that
$H=(I-\widetilde{\alpha})(Y)$ is a proper subgroup of $Y$. This implies that
$K=A(X,H)\neq\\{0\\}$.

Let us check that $Y^{(2)}\setminus H\neq\emptyset$. Suppose the contrary is
true, i.e., $Y^{(2)}\subset H$. Then, on the one hand, for $H$ we have the
following possibilities:

$H=\\{(2m,2n):m,n\in\mathbb{Z}\\},\quad H=\\{(2m,n):m,n\in\mathbb{Z}\\},$ (31)
$H=\\{(m,2n):m,n\in\mathbb{Z}\\},\quad
H=\\{(2m,2n),(2m-1,2n-1):m,n\in\mathbb{Z}\\}.$ (32)

On the other hand, $H=\\{((1-a)m-bn,-cm+(1-d)n):m,n\in\mathbb{Z}\\}$ and this
implies that

$(-b,1-d),(1-a,-c)\in H.$ (33)

Since $a+d=-3$, this implies that $a$ and $d$ have different parity. In view
of $ad-bc=1$, we have either $a$ is even and $b,c,d$ are odd or $d$ is even
and $a,b,c$ are odd. Anyway (33) contradicts (31) and (32). Hence
$Y^{(2)}\setminus H\neq\emptyset$.

Take $0<\kappa<1$ and consider on the group $X$ the distribution

$\pi_{1}=\kappa E_{(1,1)}+(1-\kappa)m_{K}$

and the signed measure

$\pi_{2}=\frac{1}{\kappa}E_{(1,1)}+\frac{\kappa-1}{\kappa}m_{K}.$

Since $H=A(Y,K)$, it follows from (4) that

$\widehat{m}_{K}(y)=\begin{cases}1,&\text{\ if\ }\ y\in H,\\\ 0,&\text{\ if\
}\ y\not\in H,\end{cases}$

and the characteristic functions $\hat{\pi}_{j}(y)$ are of the form

$\hat{\pi}_{1}(y)=\begin{cases}1,&\text{\ if\ }\ y\in H,\\\ \kappa,&\text{\
if\ }\ y\not\in
H,\end{cases}\quad\quad\hat{\pi}_{2}(y)=\begin{cases}1,&\text{\ if\ }\ y\in
H,\\\ \displaystyle{\frac{1}{\kappa}},&\text{\ if\ }\ y\not\in H.\end{cases}$
(34)

Let us check that the characteristic functions $\hat{\pi}_{j}(y)$ satisfy
equation (30). To see this we verify that for any $u,v\in Y$ both sides of
equation (30) are equal to 1. Suppose that there exist $u,v\in Y$ such that
$\hat{\pi}_{1}(u+v)\hat{\pi}_{2}(u+\widetilde{\alpha}v)\neq 1$. In view of
(34) then either $u+v\in H,u+\widetilde{\alpha}v\not\in H$ or $u+v\not\in
H,u+\widetilde{\alpha}v\in H$. In both cases we obtain that
$(I-\widetilde{\alpha})v\not\in H$. This is impossible because
$H=(I-\widetilde{\alpha})(Y)$. Thus, the left hand-side of equation (30) for
any $u,v\in Y$ is equal to 1. Reasoning similarly we check that the right
hand-side of equation (30) for any $u,v\in Y$ is also equal 1. Hence the
characteristic functions $\hat{\pi}_{j}(y)$ satisfy equation (30).

It is easy to see that since (29) is valid, the functions

$h_{j}(m,n)=\exp\\{-\langle A_{j}(m,n),(m,n)\rangle\\},\quad j=1,2,$

satisfy equation (30). Hence the functions

$g_{j}(m,n)=h_{j}^{k}(m,n)\hat{\pi}_{j}(m,n),\quad(m,n)\in Y,\ j=1,2,$ (35)

for any natural $k$ also satisfy equation (30).

It follows from (28) that there is $\varepsilon>0$ such that

$\langle A_{j}(m,n),(m,n)\rangle\geq\varepsilon(m^{2}+n^{2}),\quad j=1,2.$
(36)

Since $g_{j}(0,0)=1$, inequalities (36) imply that for a big enough $k$ the
inequalities

$\sum_{(m,n)\in Y}g_{j}(m,n)<2,\quad j=1,2,$ (37)

hold true. Put

$\rho_{j}(z,w)=\sum_{(m,n)\in Y}g_{j}(m,n)\bar{z}^{m}\bar{w}^{n},\quad(z,w)\in
X,\ j=1,2.$

Since $g_{j}(-y)=g_{j}(y)$ for all $y\in Y$, it follows from (37) that then
$\rho_{j}(z,w)>0$ for all $(z,w)\in X$, $j=1,2$. It is also obvious that

$\int\limits_{X}\rho_{j}(z,w)dm_{X}(z,w)=1,\quad j=1,2.$

Thus, the functions $\rho_{j}(z,w)$ are densities with respect to $m_{X}$ of
some distributions $\mu_{j}$ on the group $X$. In so doing,
$\hat{\mu}_{j}(y)=g_{j}(y)$, $j=1,2$. Let $\xi_{1}$ and $\xi_{2}$ are
independent random variables with values in the group $X$ and distributions
$\mu_{1}$ and $\mu_{2}$. Since the characteristic functions $\hat{\mu}_{j}(y)$
satisfy equation (30), by Lemma 3.3, the conditional distribution of the
linear form $L_{2}=\xi_{1}+\alpha\xi_{2}$ given $L_{1}=\xi_{1}+\xi_{2}$ is
symmetric. Note now that $Y^{(2)}=A(Y,G)$. For this reason if $\mu\in{\rm
M}^{1}(X)$ and $\mu\in\Gamma(X)*{\rm M}^{1}(G)$, then the restriction of the
characteristic function $\hat{\mu}(y)$ to the subgroup $Y^{(2)}$ is the
characteristic function of a Gaussian distribution. Since $Y^{(2)}\setminus
H\neq\emptyset$, it follows from (34) and (35) that
$\mu_{j}\notin\Gamma(X)*{\rm M}^{1}(G)$, $j=1,2$. $\Box$

In view of Theorem 3.1, it is interesting to note that Theorem A holds true
for the group ${\mathbb{T}}^{2}$ if we substitute condition $(\ref{d1})$ for
$(\ref{fe18})$ ([4], see also [6, Theorem 16.8]). Namely, the following
statement holds.

######  Theorem B

Consider the $2$-dimensional torus $\mathbb{T}^{2}$ and let $G$ be the
subgroup of $\mathbb{T}^{2}$ generated by all elements of $\mathbb{T}^{2}$ of
order $2$. Assume that a topological automorphism $\alpha$ of the group
$\mathbb{T}^{2}$ satisfies conditions $(\ref{fe18})$. Let $\xi_{1}$ and
$\xi_{2}$ be independent random variables with values in the group
$\mathbb{T}^{2}$ and distributions $\mu_{1}$ and $\mu_{2}$ with nonvanishing
characteristic functions. If the conditional distribution of the linear form
$L_{2}=\xi_{1}+\alpha\xi_{2}$ given $L_{1}=\xi_{1}+\xi_{2}$ is symmetric, then
$\mu_{j}=\gamma_{j}*\omega_{j}$, where $\gamma_{j}\in\Gamma(\mathbb{T}^{2})$,
$\omega_{j}\in{\rm M^{1}}(G)$, $j=1,2$. In so doing the Gaussian distributions
$\gamma_{j}$ are concentrated on cosets of the same dense one-parameter
subgroup in $\mathbb{T}^{2}$.

In conclusion, taking into account Theorems 2.1 and 3.1, we formulate the
following conjecture.

######  Conjecture 3.4

Let $X$ be a second countable locally compact Abelian group, let $G$ be the
subgroup of $X$ generated by all elements of $X$ of order $2$, and let
$\alpha$ be a topological automorphism of the group $X$ satisfying condition
$(\ref{d1})$. Let $\xi_{1}$ and $\xi_{2}$ be independent random variables with
values in $X$ and distributions $\mu_{1}$ and $\mu_{2}$ with nonvanishing
characteristic functions. The symmetry of the conditional distribution of the
linear form $L_{2}=\xi_{1}+\alpha\xi_{2}$ given $L_{1}=\xi_{1}+\xi_{2}$
implies that $\mu_{j}\in\Gamma(X)*{\rm M^{1}}(G)$, $j=1,2$, if and only if the
group $X$ contains no subgroups topologically isomorphic to the
$2$-dimensional torus $\mathbb{T}^{2}$.

The sufficiency in this assertion follows from Theorem 2.1.

Acknowledgements This article was written during my stay at the Department of
Mathematics University of Toronto as a Visiting Professor. I am very grateful
to Ilia Binder for his invitation and support. I would also like to thank the
reviewer for carefully reading the article and useful comments.

## References

  * [1] Bogachev, V.I.: Gaussian Measures, Mathematical Surveys and Monographs, 62. American Mathematical Society, Providence, RI (1998)
  * [2] Feldman, G.M.: Gaussian distributions on locally compact Abelian groups. Theory Probab. Appl. 23, 529–542 (1979). https://doi.org/10.1137/1123062
  * [3] Feldman, G.M.: On the Heyde theorem for finite Abelian groups. J. Theoretical Probab. 17, 929–941 (2004). https://doi.org/10.1007/s10959-004-0583-0
  * [4] Feldman, G.M.: On a characterization theorem for locally compact abelian groups. Probab. Theory Relat. Fields. 133, 345–357 (2005). https://doi.org/10.1007/s11118-020-09885-x
  * [5] Feldman, G.M.: On the Heyde theorem for discrete Abelian groups. Studia Math. 177, 67–79 (2006). https://doi.org/10.4064/sm177-1-5
  * [6] Feldman, G.: Functional Equations and Characterization Problems on Locally Compact Abelian Groups, EMS Tracts in Mathematics, 5. European Mathematical Society, Zürich, (2008)
  * [7] Feldman, G.M.: The Heyde theorem for locally compact Abelian groups. J. Funct. Anal.  258, 3977–3987 (2010). https://doi.org/10.1016/j.jfa.2010.03.005
  * [8] Feldman, G.M.: On a characterization of convolutions of Gaussian and Haar distributions. Math. Nachr. 286, 340–348 (2013). https://doi.org/10.1002/mana.201100320
  * [9] Feldman, G.M.: On a characterization theorem for the group of $p$-adic numbers. Publ. Math. Debrecen. 87, 147–166 (2015). https://doi.org/10.5486/PMD.2015.7100
  * [10] Feldman, G.M.: Heyde’s characterization theorem on some locally compact Abelian groups. Theory Probab. Appl.  62, 399–412 (2018). https://doi.org/10.1137/S0040585X97T988708
  * [11] Feldman, G.M.: On a characterization theorem for probability distributions on discrete Abelian groups. Theory Probab. Appl.  63, 594–612 (2019). https://doi.org/10.1137/S0040585X97T989271
  * [12] Feldman, G.M.: On a characterization theorem for connected locally compact Abelian groups. J. Fourier Anal. Appl.  26, no. 1, Paper No. 14, 22 pp., (2020). https://doi.org/10.1007/s00041-019-09721-w
  * [13] Feldman, G.M.: A characterization theorem on compact Abelian groups. J. Fourier Anal. Appl.  27, no. 5, Paper No. 86, 23 pp., (2021). https://doi.org/10.1007/s00041-021-09889-0
  * [14] Feldman, G.: Characterization of Probability Distributions on Locally Compact Abelian Groups, Mathematical Surveys and Monographs, 273. American Mathematical Society, Providence, RI (2023)
  * [15] Feldman, G.M.: Heyde theorem on locally compact Abelian groups with the connected component of zero of dimension 1. Potential Analysis, (2023). https://doi.org/10.1007/s11118-023-10095-4
  * [16] Fuchs, L: Infinite Abelian Groups, 2, Pure Applied Math, 36-II. Academic Press, New York (1973)
  * [17] Hewitt, E., Ross, K. A.: Abstract Harmonic Analysis, 1: Structure of Topological Groups, Integration Theory, Group Representations, 2nd ed., Grundlehren der Mathematischen Wissenschaften, 115, Springer-Verlag, New York (1994)
  * [18] Heyde, C.C.: Characterization of the normal low by the symmetry of a certain conditional distribution. Sankhya. 32, Ser. A. 115–118 (1970). http://www.jstor.org/stable/25049642
  * [19] Kagan, A. M., Linnik, Yu. V., Rao C.R.: Characterization Problems in Mathematical Statistics, Wiley Series in Probability and Mathematical Statistics. John Wiley $\&$ Sons, New York (1973)
  * [20] Myronyuk, M.V.: Heyde’s characterization theorem for discrete abelian groups. J. Aust. Math. Soc. 88, 93–102 (2010). https://doi.org/10.1017/S1446788709000378
  * [21] Myronyuk, M.V.: The Heyde theorem on a-adic solenoids. Colloquium Mathematicum. 132, 195–210 (2013) https://doi.org/10.4064/cm132-2-3
  * [22] Myronyuk, M.V., Feldman, G.M.: On a characterization theorem on finite abelian groups. Siberian Math. J. 46, 315–324 (2005). https://doi.org/10.1007/s11202-005-0033-y
  * [23] Myronyuk, M.V., Feldman, G.M.: Independent linear statistics on the two-dimensional torus. Theory Probab. Appl. 52, 78–92 (2008). https://doi.org/10.1137/S0040585X97982785
  * [24] Parthasarathy, K.R.: Probability Measures on Metric Spaces. Academic Press, New York (1967)

B. Verkin Institute for Low Temperature Physics and Engineering
of the National Academy of Sciences of Ukraine
Nauky Ave. 47, 61103 Kharkiv, Ukraine

Department of Mathematics University of Toronto
40 St. George Street Toronto, ON, M5S 2E4 Canada

e-mail<EMAIL_ADDRESS>


Version of August 28, 2024$,\,\,\,$

# Set-Valued Analysis of Generalized Barycentric Coordinates and Their
Geometric Properties

Fabio V. Difonzo  Istituto per le Applicazioni del Calcolo “Mauro Picone”,
Consiglio Nazionale delle Ricerche, Via G. Amendola 122/I, 70126 Bari, Italy
<EMAIL_ADDRESS>
###### Abstract.

Letting $P$ be a convex polytope in $\mathbb{R}^{d}$ with $n>d$ vertices, we
study geometric and analytical properties of the set of generalized
barycentric coordinates relative to any point $p\in P$. We prove that such
sets are polytopes in $\mathbb{R}^{n}$ with at most $n-d-1$ vertices, and
provide results about continuity and differentiability for the corresponding
set-valued maps.

###### Key words and phrases:

Generalized Barycentric Coordinates; Convex Polytopes; Set-Valued Maps

###### 1991 Mathematics Subject Classification:

52A21, 26E25

##  1\. Introduction

Barycentric coordinates, first introduced by Möbius [13], are nowadays useful
tools in interpolation problems [10, 9], computational mechanics [4], computer
graphics [15] and differential equation theory [6]. We refer to [12] and all
the references therein for a thorough source of theory and applications of
generalized barycentric coordinates. In this work we attempt to study them as
set-valued maps from convex $d$-polytopes with $n$ vertices, $n>d$, having
values onto the unit cube $[0,1]^{n}$, and to provide some fundamental
geometrical and analytical properties about them.

##  2\. Setting

Let $n,d\in\mathbb{N}$, with $n>d$,
$\\{v_{i}\\}_{i=1}^{n}\subseteq\mathbb{R}^{d}$ be a set of vertices and let

$V\mathrel{\mathop{:}}=\begin{bmatrix}v_{1}&\cdots&v_{n}\end{bmatrix}\in\mathbb{R}^{n\times
d}.$

Defining $\mathds{1}$ to be the vector whose components are all equal to one,
and assuming that $\dim\ker\begin{bmatrix}V\\\
\mathds{1}^{\top}\end{bmatrix}=n-d-1$, then there exists
$N\in\mathbb{R}^{n\times(n-d-1)}$ such that

(2.1) $\langle N\rangle=\ker\begin{bmatrix}V\\\
\mathds{1}^{\top}\end{bmatrix}.$

Let $P\mathrel{\mathop{:}}=\mathrm{conv}V$ be the convex polytope whose
vertices are given by the columns of $V$.

######  Definition 2.1.

Let $p\in P$. A vector $\lambda(p)\in\mathbb{R}^{n}$ is a set of generalized
barycentric coordinates for $p$ if it is a componentwise nonnegative solution
to the linear system

(2.2) $\begin{bmatrix}V\\\
\mathds{1}^{\top}\end{bmatrix}\lambda=\begin{bmatrix}p\\\ 1\end{bmatrix}.$

When $\lambda\geq 0$ componentwise, we will also say that $\lambda$ is
feasible. It comes from the definition that, for each $p\in P$ there always
exists a set of feasible generalized barycentric coordinates $\tau(p)\geq 0$.
More precisely, since (2.2) is full rank, it is meaningful to define the set,
depending on $p$,

(2.3)
$\Gamma(p)\mathrel{\mathop{:}}=\\{c\in\mathbb{R}^{n-d-1}\,:\,\tau(p)+Nc\geq
0\\},\quad p\in P.$

Thus, we can consider the following set-valued map

$\displaystyle\Gamma:P$ $\displaystyle\rightrightarrows\mathbb{R}^{n-d-1}$
$\displaystyle p$ $\displaystyle\mapsto\Gamma(p)$

It then follows that the entire set of feasible barycentric coordinates
relative to $p\in P$ can be expressed by the set-valued map

(2.4) $\displaystyle\Lambda:P$ $\displaystyle\rightrightarrows[0,1]^{n}$
$\displaystyle p$ $\displaystyle\mapsto\tau(p)+N\Gamma(p).$

We prove that $\Lambda(p)$, and hence $\Gamma(p)$, do not depend on the
particular $\tau(p)$ chosen. In fact, if $\tau^{\prime}(p)$ is another
feasible set of barycentric coordinates for $p$, then there exists
$c^{\prime}\in\Gamma(p)$ such that

$\tau(p)=\tau^{\prime}(p)+Nc^{\prime},$

and therefore
$\Lambda(p)=\tau^{\prime}(p)+Nc^{\prime}+N\Gamma(p)=\tau^{\prime}(p)+N\Gamma(p)$.
Let us also notice that for each $c\in\Gamma(p)$ there exists a unique
$\lambda\in\Lambda(p)$ such that $\lambda(p)=\tau(p)+Nc$ and, thus, we have a
bijective correspondence between the two sets $\Gamma(p)$ and $\Lambda(p)$.

It is easily seen that, from (2.2),

$\Lambda(p)=\left\\{\lambda\in\mathbb{R}^{n}\,:\,\begin{bmatrix}V\\\
\mathds{1}^{\top}\end{bmatrix}\lambda=\begin{bmatrix}p\\\
1\end{bmatrix},\,\lambda\geq 0\right\\},$

where, hereafter, $\lambda\geq 0$ is meant to hold componentwise.
This paper is devoted to study geometric and analytical properties of the set-
valued maps $\Gamma(p)$ and $\Lambda(p)$.
In Section 3 we prove that $\Lambda(p)$ is a bounded polytope, and we describe
its vertices. In Section 4 we study smoothness of these maps.

##  3\. Geometric structure of $\Lambda(p)$ and $\Gamma(p)$

In this section we state and prove fundamental results about the geometric
features of $\Lambda(p)$ and $\Gamma(p)$. Such results will be useful for
investigating analytical properties of the set-valued maps introduced above.

It holds the following.

######  Proposition 3.1.

For each $p\in P$, $\Gamma(p)$ and $\Lambda(p)$ are polytopes of dimension at
most $n-d-1$.

###### Proof.

From (2.3) $\Gamma(p)$ is a closed convex polyhedron and, since $\tau(p)$ can
always be chosen such that $\tau(p)\geq 0$, $0\in\Gamma(p)$, therefore
providing $\Gamma(p)^{\circ\circ}=\Gamma(p)$, where $\Gamma(p)^{\circ}$ is the
polar of $\Gamma(p)$. From classical results (e.g., see [3, 11]),
$\Gamma(p)^{\circ}$ is a polyhedron, so that $\Gamma(p)^{\circ\circ}$ is a
polytope, and so is $\Gamma(p)$. Since $\Lambda(p)=\tau(p)+N\Gamma(p)$, then
$\Lambda(p)$ is a polytope as well.
Thus $\Gamma(p)$ is a convex, and hence affine, subspace of
$\mathbb{R}^{n-d-1}$, and so $\dim\Gamma(p)\leq n-d-1$. By Caratheodory’s
Theorem ([3]) let $\\{c_{1},\ldots,c_{n-d}\\}$ be an affine base for
$\Gamma(p)$.

Given an arbitrary $\lambda\in\Lambda(p)$, it can be written as
$\lambda=\tau+Nc$ for some unique $c\in\Gamma(p)$. Therefore, there exist
$\alpha_{i}\geq 0,i=1,\ldots,n-d$, such that

$c=\sum_{i=1}^{n-d}\alpha_{i}c_{i},\quad\sum_{i=1}^{n-d}\alpha_{i}=1.$

Letting $\lambda_{i}\mathrel{\mathop{:}}=\tau+Nc_{i},i=1,\ldots,n-d$, it
follows that $\lambda_{i}\in\Lambda(p)$ and

$\lambda=\sum_{i=1}^{n-d}\alpha_{i}\lambda_{i}.$

Thus $\Lambda(p)\subseteq\mathrm{conv}\\{\lambda_{i}\,:\,i=1,\ldots,n-d\\}$,
implying $\dim\Lambda(p)\leq n-d-1$. ∎

######  Remark 3.2.

If $p\in\partial P$ and since facet reducing property is shared by all
feasible barycentric coordinates, then $\dim\Lambda(p)=0$.
If $p\in\mathrm{int}P$ then, by Caratheodory’s Theorem, every
$\lambda\in\Lambda(p)$ can be written as the convex combination of $n-d$
elements in $\Lambda(p)$.

######  Corollary 3.3.

$\Gamma:P\rightrightarrows\mathbb{R}^{n-d-1}$ and
$\Lambda:P\rightrightarrows[0,1]^{n}$ have compact convex values.

###### Proof.

It is a direct consequence of Proposition 3.1. ∎

Next, we want to describe the vertices of the $\Lambda(p)$ as a polytope. We
prove that they are given by the simplicial barycentric coordinates that are
feasible at $p$, as defined below. In order to prove this, we introduce some
notation.

For each $i=1,\ldots,n$, let

$\mathcal{I}_{i}\mathrel{\mathop{:}}=\\{i,i+1,\ldots,i+n-d-2\\},$

where indices are considered circularly (i.e., $n+1\equiv n$ and so on). Let
us observe that $|\mathcal{I}_{i}|=n-d-1$. Let then $\sigma^{i}(p)$ be the
unique solution to

(3.1) $\begin{bmatrix}V\\\ \mathds{1}^{\top}\\\
I_{i}\end{bmatrix}\sigma=\begin{bmatrix}p\\\ 1\\\ 0\end{bmatrix},$

where $I_{i}$ is a diagonal matrix such that

$I_{i}(j,j)\mathrel{\mathop{:}}=\begin{cases}1,&j\in\mathcal{I}_{i},\\\
0,&j\notin\mathcal{I}_{i}.\end{cases}$

As $\\{v_{i}\,:\,i=1,\ldots,n\\}$ are such that $\dim\ker\begin{bmatrix}V\\\
\mathds{1}^{\top}\end{bmatrix}=n-d-1$, it follows that $\begin{bmatrix}V\\\
\mathds{1}^{\top}\\\ I_{i}\end{bmatrix}$ is invertible, and thus (3.1) has a
unique solution $\sigma^{i}(p)$, which we refer to as _simplicial barycentric
coordinates_ for $p$. It is straightforward to realize that

$\displaystyle\sigma^{i}_{j}(p)$ $\displaystyle=0,\,\,\textrm{ if
}j\in\mathcal{I}_{i},$ $\displaystyle\sigma^{i}_{j}(p)$ $\displaystyle\neq
0,\,\,\textrm{ if }j\notin\mathcal{I}_{i}.$

It holds the following

######  Theorem 3.4.

Let $p\in P$. Then there exists $I(p)\subseteq\\{1,\ldots,n\\}$ such that
$|I(p)|=n-d$ and $\sigma^{i}(p)\geq 0$ is a vertex of $\Lambda(p)$ for all
$i\in I(p)$. Therefore

$\Lambda(p)=\mathrm{conv}\\{\sigma^{i}(p)\,:\,i\in I(p)\\}.$

###### Proof.

From Proposition 3.1, it suffices to prove that there exist at most, and thus
exactly by Caratheodory’s Theorem, $n-d$ feasible simplicial barycentric
coordinates $\sigma^{i}(p)$ at $p$. Proceeding by contradiction, let us
suppose that for any $\overline{n}>n-d$ and for any $i=1,\ldots,\overline{n}$,
there is some $j_{i}=1,\ldots,n$ such that $\sigma^{i}_{j_{i}}(p)<0$, where
$\sigma^{i}(p)$ is the simplicial barycentric coordinates for $p$ defined in
(3.1). This implies that

$p\notin
S_{i}\mathrel{\mathop{:}}=\mathrm{conv}\\{v_{j}\,:\,j\notin\mathcal{I}_{i}\\},$

for all $i=1,\ldots,\overline{n}$. Let us note that each $S_{i}$ is a
$d$-simplex. This contradicts the fact that the polytope $P$ has at least
$n-d$ simplices (see [8, 11, 16]).
Let then $I(p)\subseteq\\{1,\ldots,n\\}$ be such that $|I(p)|=n-d$ and
$\sigma^{i}(p)\geq 0$ for all $i\in I(p)$, and let us now prove that
$\sigma^{i}(p)$ are vertices of $\Lambda(p)$. In fact, let $i\in I(p)$ be
given and let $\lambda,\mu\in\Lambda(p)$ and $a\in(0,1)$ be such that

$\sigma^{i}(p)=(1-a)\lambda+a\mu.$

Therefore, for all $j\in\mathcal{I}_{i}$,

$0=(1-a)\lambda_{j}+a\mu_{j},$

so that, by nonnegativity, it follows that $\lambda_{j}=\mu_{j}=0$, and thus
$\lambda,\mu$ both solve (3.1). Since it is nonsingular, we deduce
$\lambda=\mu=\sigma^{i}(p)$, implying that $\sigma^{i}(p)$ is a vertex of
$\Lambda(p)$. Resorting to Krein-Milman Theorem proves the claim. ∎

######  Example 3.5.

Let us consider a convex quadrilateral $Q\subseteq\mathbb{R}^{2}$ defined as

$Q=\mathrm{conv}\\{v_{i}\,:\,i=1,2,3,4\\},$

where the $v_{i}$’s are assumed to be in general position. In this case,
letting $p\in Q$, Theorem 3.4 implies that $\Lambda(p)$ is a segment in
$\mathbb{R}^{4}$, whose vertices are given by the triangular barycentric
coordinates nonnegative at $p$. In other words, assuming without loss of
generality that
$p\in\mathrm{conv}\\{v_{1},v_{2},v_{3}\\}\cap\mathrm{conv}\\{v_{1},v_{2},v_{4}\\}$,
and letting $\tau^{4}(p),\tau^{3}(p)$ be the unique solutions to

$\begin{bmatrix}V\\\ \mathds{1}^{\top}\\\
\begin{matrix}0&0&0&1\end{matrix}\end{bmatrix}\tau^{4}(p)=\begin{bmatrix}p\\\
1\\\ 0\end{bmatrix},\begin{bmatrix}V\\\ \mathds{1}^{\top}\\\
\begin{matrix}0&0&1&0\end{matrix}\end{bmatrix}\tau^{3}(p)=\begin{bmatrix}p\\\
1\\\ 0\end{bmatrix}$

respectively, then $\Lambda(p)=\mathrm{conv}\\{\tau^{3}(p),\tau^{4}(p)\\}$.

In fact, here $N\in\mathbb{R}^{4}$ and it is easily seen that for any
$\lambda(p)\in\Lambda(p)$ there exists a unique $c\in[a,b]$ such that

$\lambda(p)=\tau^{4}(p)+cN,$

where we define
$a\mathrel{\mathop{:}}=\max\left\\{-\frac{\tau^{4}_{i}(p)}{N_{i}}\,:\,N_{i}>0\right\\},b\mathrel{\mathop{:}}=\min\left\\{-\frac{\tau^{4}_{i}(p)}{N_{i}}\,:\,N_{i}<0\right\\}$
(see [7]). It is completely evident that one between $a,b$ must be zero:
assuming it to be $a$, then $b>0$; thus $c=a$ provides
$\lambda(p)=\tau^{4}(p)$, while $c=b$ provides $\lambda(p)=\tau^{3}(p)$.
Therefore, since $\tau^{4}(p)-\tau^{3}(p)\in\langle N\rangle$, we deduce that
$\lambda(p)\in\mathrm{conv}\\{\tau^{3}(p),\tau^{4}(p)\\}$, as expected. If
$b=0$, then it is enough, without loss of generality, to consider $-N$ instead
of $N$ in the arguments above.

######  Example 3.6.

For pentagons or polytopes with more than five vertices in $\mathbb{R}^{2}$, a
similar idea has been explored in [1], where authors propose to consider a
suitably blended average of barycentric coordinates relative to a given
triangularization of the polytope. As a matter of fact, Theorem 3.4 provides a
theoretical justification to the blending approach.

More explicitly, let us consider a convex pentagon $P\subseteq\mathbb{R}^{2}$
defined as

$P\mathrel{\mathop{:}}=\mathrm{conv}\\{v_{i}\,:\,i=1,\ldots,5\\},$

with $\dim\ker\begin{bmatrix}V\\\ \mathds{1}^{\top}\end{bmatrix}=2$. Let now
$\mathcal{T}_{ijk}\mathrel{\mathop{:}}=\mathrm{conv}\\{v_{l}\,:\,l=i,j,k\\}$
and $\tau_{ijk}(p)$ be its corresponding triangular barycentric coordinates
relative to $p\in\mathcal{T}_{123}\cap\mathcal{P}_{124}\cap\mathcal{P}_{125}$.
Thus, $\tau_{123},\tau_{124},\tau_{125}$ are, respectively, the unique
solutions to

$\begin{bmatrix}V\\\ \mathds{1}^{\top}\\\
\begin{matrix}0&0&0&1&0\end{matrix}\\\
\begin{matrix}0&0&0&0&1\end{matrix}\end{bmatrix}\tau=\begin{bmatrix}p\\\ 1\\\
0\\\ 0\end{bmatrix},\begin{bmatrix}V\\\ \mathds{1}^{\top}\\\
\begin{matrix}0&0&1&0&0\end{matrix}\\\
\begin{matrix}0&0&0&0&1\end{matrix}\end{bmatrix}\tau=\begin{bmatrix}p\\\ 1\\\
0\\\ 0\end{bmatrix},\begin{bmatrix}V\\\ \mathds{1}^{\top}\\\
\begin{matrix}0&0&1&0&0\end{matrix}\\\
\begin{matrix}0&0&0&1&0\end{matrix}\end{bmatrix}\tau=\begin{bmatrix}p\\\ 1\\\
0\\\ 0\end{bmatrix}.$

It is easy to see that $\tau_{123},\tau_{124},\tau_{125}$ are linearly
independent and $\tau_{124}(p)-\tau_{123}(p),\tau_{125}-\tau_{123}$ span
$\ker\begin{bmatrix}V\\\ \mathds{1}^{\top}\end{bmatrix}$. Thus, every feasible
generalized barycentric coordinates at $p$, say $\lambda(p)$, can be written
as

$\lambda(p)=\tau_{123}(p)+c(\tau_{124}(p)-\tau_{123}(p))+\overline{c}(\tau_{125}(p)-\tau_{123}(p)),$

for some $c,\overline{c}\in\mathbb{R}$, which implies

$\lambda(p)=(1-c-\overline{c})\tau_{123}(p)+c\tau_{124}(p)+\overline{c}\tau_{125}(p).$

However, since $\lambda(p)\geq 0$ component-wise and because of the structure
of $\tau_{ijk}(p)$, we get that $c,\overline{c}\in[0,1]$, so that $\lambda(p)$
is a convex combination of $\tau_{123},\tau_{234},\tau_{345}$. Therefore
$\Lambda(p)=\mathrm{conv}\\{\tau_{123},\tau_{234},\tau_{345}\\}$, which
configures a triangle in $\mathbb{R}^{5}$, thus confirming Theorem 3.4.

######  Example 3.7.

Let us consider a pyramid $P$ in $\mathbb{R}^{3}$ defined as

$P\mathrel{\mathop{:}}=\mathrm{conv}\\{v_{i}\,:\,i=1,\ldots,5\\},$

with $\dim\ker\begin{bmatrix}V\\\ \mathds{1}^{\top}\end{bmatrix}=1$. This case
is analogous to the quadrilateral in Example 3.5, and again Theorem 3.4 holds
straightforwardly.
Let us now consider a prism $P$ in $\mathbb{R}^{3}$ defined as

$P\mathrel{\mathop{:}}=\mathrm{conv}\\{v_{i}\,:\,i=1,\ldots,8\\},$

with $\dim\ker\begin{bmatrix}V\\\ \mathds{1}^{\top}\end{bmatrix}=4$. In this
case, it is immediate to observe that the barycentric coordinates
$\sigma_{ijkl}$ relative to the simplices
$\mathrm{conv}\\{v_{m}\,:\,m=i,j,k,l\\}$, with $ijkl\in
I\mathrel{\mathop{:}}=\\{1235,1236,1237,1238,1245\\}$, span
$\ker\begin{bmatrix}V\\\ \mathds{1}^{\top}\end{bmatrix}$ and thus, again
resorting to arguments similar as in Example 3.6 by counting and exploiting
the zero structure of such barycentric coordinates, we deduce that
$\Lambda(p)=\mathrm{conv}\\{\sigma_{ijkl}\,:\,ijkl\in I\\}$.

Let us notice that the vertices of $\Lambda(p)$ are strictly related to $p\in
P$, and thus the set $I(p)$ is expected to vary with $p$. In fact, it is a
simple observation that if $p\in\partial S_{i}$ for some $i=1,\ldots,n$, then
there must exist some $j\notin I(p)$ such that $p\in\partial S_{j}$: for such
a point, then, $\sigma^{i}(p)=\sigma^{j}(p)$.
Such a continuity of $\Lambda(p)$ is related to the selection problem, i.e.
all the possible sets of generalized barycentric coordinates. Because of
(2.4), continuity of $\Gamma(p)$ and $\Lambda(p)$ imply each other, provided
that $\tau(p)$ in (2.4) is smooth enough. Since $\Lambda(p)$ does not depend
on the particular $\tau(p)$ chosen and simplicial barycentric coordinates are
always smooth and feasible as long as $p$ belongs to their corresponding
simplices, we will always assume $\tau(p)$ to be as smooth as needed. We are
going to further deepen this topic in the next section.

##  4\. Analytical Properties of $\Lambda(p)$ and $\Gamma(p)$

In this section we investigate smoothness of the generalized barycentric
coordinates maps.

The following results describe some analytical properties of $\Gamma(p)$ and
$\Lambda(p)$.

######  Proposition 4.1.

$\Lambda:P\rightrightarrows[0,1]^{n}$ is upper semi-continuous.

###### Proof.

Since $P$ is closed and $[0,1]^{n}$ is compact, we only need to prove that
$\mathrm{Graph}(\Lambda)$ is closed (see [2]). Letting
$(p,\lambda)\in\overline{\mathrm{Graph}(\Lambda)}$, it follows that there
exist $\\{p_{k}\\}\subseteq P,\\{\lambda_{k}\\}\subseteq[0,1]$ such that

$\lim_{k\to\infty}p_{k}=p,\,\,\lim_{k\to\infty}\lambda_{k}=\lambda,$

and

$\lambda_{k}\in\Lambda(p_{k}),\,\,k\in\mathbb{N}.$

Therefore $V\lambda_{k}=p_{k}$ for all $k\in\mathbb{N}$ and so, taking the
limit of both sides, $V\lambda=p$, implying $\lambda\in\Lambda(p)$. It then
holds that $\mathrm{Graph}(\Lambda)$ is closed and thus $\Lambda$ is upper
semi-continuous. ∎

######  Proposition 4.2.

$\Lambda:P\rightrightarrows[0,1]^{n}$ is lower semi-continuous.

###### Proof.

Let $p\in P$ and $\\{p_{k}\\}\in P$ such that $p_{k}\to p$ as $k\to\infty$;
let also $\lambda\in\Lambda(p)$. Therefore, for each $k\in\mathbb{N}$ there
exists $\lambda_{k}\in[0,1]^{n}$ such that $p_{k}=V\lambda_{k}$ and
$\mathds{1}^{\top}\lambda_{k}=1$. Thus

$\lim_{k\to\infty}V\lambda_{k}=V\lambda,$

and so, since $V$ is full column rank, it follows from standard arguments that
$\\{\lambda_{k}\\}$ converges and

$\lim_{k\to\infty}\lambda_{k}=\lambda.$

Since $\lambda_{k}\in\Lambda(p_{k})$ by construction, the claim is proved. ∎

It is a direct consequence of definition that $\Gamma(p)$ is also upper and
lower semi-continuous. We then have the following.

######  Corollary 4.3.

$\Lambda:P\rightrightarrows[0,1]^{n}$ and
$\Gamma:P\rightrightarrows\mathbb{R}^{n-d-1}$ are continuous set-valued maps.

###### Proof.

It is a straightforward consequence of Propositions 4.1, 4.2 and definition of
$\Gamma$. ∎

We want to focus now on smoothness features of $\Lambda(p)$. In particular, we
are going to prove that it is semidifferentiable in $P$. We recall here some
useful definitions for the next results (see [14, 5] for further details).
Let $X$ be a normed vector space and $F:X\rightrightarrows\mathbb{R}^{n}$ be a
set-valued map with nonempty images.

######  Definition 4.4.

The set-valued map $F:X\rightrightarrows\mathbb{R}^{n}$ is called
_semidifferentiable_ at a point
$(\overline{x},\overline{y})\in\mathrm{Graph}F$ if the limit

$DF(\overline{x},\overline{y};h_{0})\mathrel{\mathop{:}}=\lim_{t_{n}\to
0^{+},\,h_{n}\to
h_{0}}t_{n}^{-1}\left[F(\overline{x}+t_{n}h_{n})-\overline{y}\right]$

exists for all $h_{0}\in X$ in the sense of Kuratowski-Painlevé (see [2]).

######  Theorem 4.5.

$\Lambda:P\rightrightarrows[0,1]^{n}$ is semidifferentiable at every point
$(p,\lambda(p))\in\mathrm{Graph}F$ where $\lambda(p)$ is differentiable.

###### Proof.

Let $p\in P$ and $h\in P$ be given. It is known [14] that in order to prove
that $\Lambda$ is semidifferentiable we need to prove that
$D\Lambda(p,\lambda(p);h)$ is nonempty, which requires that there exists
$v\in\mathbb{R}^{n}$ such that

(4.1)
$\liminf_{(t_{n},h_{n})\to(0^{+},h)}t_{n}^{-1}d(\lambda(p)+t_{n}v,\Lambda(p+t_{n}h_{n}))=0,$

where $d(a_{0},A)\mathrel{\mathop{:}}=\inf_{a\in A}\|a_{0}-a\|$. By
definition, (4.1) is equivalent to

$\liminf_{(t_{n},h_{n})\to(0^{+},h)}t_{n}^{-1}d(\lambda(p)+t_{n}v,\lambda(p+t_{n}h_{n})+N\Gamma(p+t_{n}h_{n}))=0,$

and thus

$\liminf_{(t_{n},h_{n})\to(0^{+},h)}t_{n}^{-1}d(\lambda(p)-\lambda(p+t_{n}h_{n})+t_{n}v,N\Gamma(p+t_{n}h_{n}))=0.$

Using homogeneity and continuity of the distance function and since
$\lambda(p)$ is differentiable, we have to prove that

(4.2)
$\liminf_{(t_{n},h_{n})\to(0^{+},h)}d(-J\lambda(p)h_{n}+v,t_{n}^{-1}N\Gamma(p+t_{n}h_{n}))=0.$

Since, from Proposition 3.1, $\Gamma(p+t_{n}h_{n})$ is a bounded polytope
containing the origin for all $n\in\mathbb{N}$, then $0_{\mathbb{R}^{n}}\in
t_{n}^{-1}N\Gamma(p+t_{n}h_{n})$ for all $n\in\mathbb{N}$. Thus, for any given
$\varepsilon>0$ sufficiently small we have

$B_{\varepsilon}(0)\subseteq t_{n}^{-1}N\Gamma(p+t_{n}h_{n}).$

Let now $J\lambda(p)$ be the Jacobian of $\lambda$ at $p$. Since

$\lim_{n\to\infty}J\lambda(p)h_{n}=J\lambda(p)h,$

then there exists $\overline{n}\in\mathbb{N}$ such that for all
$n\geq\overline{n}$ it holds
$\|J\lambda(p)h_{n}-J\lambda(p)h\|\leq\varepsilon$. This implies that

$J\lambda(p)h-J\lambda(p)h_{n}\in B_{\varepsilon}(0)\subseteq
t_{n}^{-1}N\Gamma(p+t_{n}h_{n})$

for all $n\geq\overline{n}$, providing that (4.2) is satisfied for
$v\mathrel{\mathop{:}}=J\lambda(p)h$, and this proves the claim. ∎

##  5\. Conclusions

In this paper we have considered the set-valued map of generalized barycentric
coordinates $\Lambda(p)$ relative to a $d$-polytope $P$ with $n>d$ vertices,
and we have proved that, for any given point $p\in P$, values of $\Lambda(p)$
define a polytope of dimension at most $n-d-1$, whose vertices are represented
by $n-d$ nonnegative simplicial barycentric coordinates relative to $p$. This
result seems to be a dual transform of a convex polytope
$P\subseteq\mathbb{R}^{d}$, in the sense that as every point $p\in P$ can be
described by a convex combination of the $n$ vertices of $P$, so each of the
barycentric coordinates relative to $p$ can be described by a convex
combination of $n-d$ simplicial barycentric coordinates in $\mathbb{R}^{n}$.
Moreover, we have proved that the these maps are continuous and
semidifferentiable at points where the corresponding generalized barycentric
coordinate is differentiable.

## Acknowledgments

FVD is part of INdAM research group GNCS. He also gratefully thanks Prof. L.
Dieci and Prof. N. Sukumar for their support while writing this paper.

## References

  * [1] Dmitry Anisimov, Daniele Panozzo, and Kai Hormann.  Blended barycentric coordinates.  Computer Aided Geometric Design, 52-53:205–216, 2017.  Geometric Modeling and Processing 2017.
  * [2] J.P. Aubin and H. Frankowska.  Set-Valued Analysis.  Modern Birkhäuser Classics. Birkhäuser Boston, 2009.
  * [3] A. Barvinok.  A Course in Convexity.  American Mathematical Society, 2007.
  * [4] K.J. Bathe.  Finite Element Procedures.  Prentice Hall, 1996.
  * [5] Darinka Dentcheva.  Differentiable Selections and Castaing Representations of Multifunctions.  Journal of Mathematical Analysis and Applications, 223(2):371–396, 1998.
  * [6] L. Dieci and F. Difonzo.  A comparison of Filippov sliding vector fields in codimension 2.  Journal of Computational and Applied Mathematics, 262:161 – 179, 2014.
  * [7] L. Dieci and F. Difonzo.  The Moments Sliding Vector Field on the Intersection of Two Manifolds.  Journal of Dynamics and Differential Equations, 2015.
  * [8] Allan L. Edmonds.  Simplicial decompositions of convex polytopes.  Pi Mu Epsilon Journal, 5(3):124–128, 1970.
  * [9] M. S. Floater.  Generalized barycentric coordinates and applications.  Acta Numerica, 24:161–214, 2015.
  * [10] M.S. Floater.  Mean Value Coordinates.  Computer Aided Geometric Design, (20):19–27, 2003.
  * [11] Branko Grünbaum, Volker Kaibel, Victor Klee, and Günter M. Ziegler.  Convex polytopes.  Springer, New York, 2003.
  * [12] Kai Hormann and N Sukumar.  Generalized barycentric coordinates in computer graphics and computational mechanics.  CRC press, Boca Raton, FL, 2017.
  * [13] A.F. Möbius.  Der barycentrische Calcul.  J.A. Barth, 1827.
  * [14] Jean-Paul Penot.  Differentiability of Relations and Differential Stability of Perturbed Optimization Problems.  SIAM Journal on Control and Optimization, 22(4):529–551, 1984.
  * [15] T. Schneider, K. Hormann, and M. S. Floater.  Bijective Composite Mean Value Mappings.  In Proceedings of the Eleventh Eurographics/ACMSIGGRAPH Symposium on Geometry Processing, SGP ’13, pages 137–146. Eurographics Association, 2013.
  * [16] Günter M. Ziegler.  Lectures on polytopes.  Springer-Verlag, New York, 1995.


# Annotationsaurus: A Searchable Directory of Annotation Tools

Mariana Neves
German Federal Institute for
Risk Assessment (BfR),
German Center for the Protection
of Laboratory Animals (Bf3R),
Berlin, Germany
<EMAIL_ADDRESS>
Jurica Ševa
Ada Health GmbH,
Karl-Liebknecht-Str. 1,
10178 Berlin, Germany
<EMAIL_ADDRESS>

###### Abstract

Manual annotation of textual documents is a necessary task when constructing
benchmark corpora for training and evaluating machine learning algorithms. We
created a comprehensive directory of annotation tools that currently includes
93 tools. We analyzed the tools over a set of 31 features and implemented
simple scripts and a Web application that filters the tools based on chosen
criteria. We present two use cases using the directory and propose ideas for
its maintenance. The directory, source codes for scripts, and link to the Web
application are available at: https://github.com/mariananeves/annotation-tools

##  1 Introduction

Annotation tools are important resources for building customized corpora for
the most various purposes, ranging from document classification Baker et al.
(2016) to discourse relations Yang and Li (2018). When choosing an annotation
tool for a particular project, researchers should consider the task at hand,
the particularities of their system environment and the expertise of the
annotators. Frequently, it is necessary to experiment with tools, even with
the burden of installing them, before finding the one that best fits their
needs.

This is, however, a demanding task, given the high number of available tools.
Usually, while choosing an annotation tool, users ask for advice in forums and
mailing lists, look at previous surveys of annotations tools, e.g. Neves and
Ševa (2019), Neves and Leser (2012) and Fort (2016), and check the tools that
have been used in recently published corpora. As far as we know, there is no
comprehensive and searchable directory of annotation tools.

We recently defined a set of 31 criteria and carried out hands-on experiments
with 15 selected annotation tools from a total of 93 tools Neves and Ševa
(2019). As an extension to this survey, we now created a directory of
annotation tools, evaluated the remaining (non-selected) tools based on many
of these criteria and developed scripts and a Web application to allow
searching for tools. In the next section we describe our directory, discuss
how to search for annotation tools, and present two use cases.

##  2 Directory of annotation tools

We created a directory of annotation tools and evaluated the tools over some
desirable criteria. This work considers the tools that we previously collected
during the preparation of a survey Neves and Ševa (2019), extended with the
ones that we have recently found. Currently, the directory contains a total of
93 tools, as listed in our GitHub
repository111https://github.com/mariananeves/annotation-tools, along with
links to the corresponding publication (if any) and tool’s Web site (if
found).

###  2.1 List of attributes

We considered both the so-called requirements and criteria (hereafter called
only as criteria) that we defined in our survey Neves and Ševa (2019). There,
we defined five requirements for selecting tools for hands-on experiments and
26 criteria which we used for the evaluation. The criteria are related to the
tool’s publications, technical attributes, data format, and functional
properties (cf. Table 1). The chosen 31 criteria, including the possible
values that each criterion can take, as defined in our survey, is available in
our GitHub repository222https://github.com/mariananeves/annotation-
tools/blob/master/schema.

Criteria | Values | Filter
---|---|---
#requirements | Availability of the tool | yes,no | available
Type of installation | web-based,stand-alone,plug-in | type
Installation successful | yes,no | installable
Annotation successful | yes,no | workable
Ability to configure a schema | yes,no | schematic
- | Year of the last publication | - | -
Number of citations for publication | - | -
Number of citations for corpora | - | -
#technical | Year of the last version | - | -
Availability of source code | yes,no | source_code
Availability of online tool | yes,no | online_available
Easiness of installation | easy,medium,hard | installation
Quality of the documentation | good,poor,none | documentation
License of the tool | full,partial,none | license
Free availability | yes,partial,no | free
#data format | Format of the schema | XML,JSON,GUI,other | format_schema
Format i/o of documents | XML,JSON,TXT,other | format_documents
Format i/o of annotations | XML,JSON,TXT,other | format_annotations
#functional | Annotation with multiple labels | yes,no | multilabel
Annotation on document level | yes,partial,no | document_level
Annotation of relations | yes,partial,no | relationships
Annotation based on ontologies | yes,no | ontologies
Support for pre-annotations | yes,partial,no | preannotations
Integration with PubMed/PMC | yes,partial,no | medline_pmc
Support for full texts | yes,partial,no | full_texts
Support for partial saving | yes,partial,no | partial_save
Support for highlighting | yes,no | highlight
Support for users and teams | yes,partial,no | users_teams
Support for IAA | yes,partial,no | iaa
Annotation of private data | yes,no | data_privacy
Support for various languages | yes,partial,no | multilingual
Table 1: List of criteria, their possible values, and corresponding filters
(if available) in the Web application.

###  2.2 Evaluation of tools

Besides the evaluation for the 15 tools previously considered in Neves and
Ševa (2019), we also carried out an evaluation for the remaining 78 tools
discarded in said survey. However, given the large number of tools, our
evaluation was restricted to the tools’ publication and documentation, i.e.,
we could not perform hands-on experiment with all of mentioned tools.

We present each tool’s evaluation (cf. “tools” folder) in a semi-structured
text file (evaluation file) for each tool. This makes the evaluation file both
human- and machine-readable. When evaluating a tool, we created a copy of the
“schema” file and checked each criterion by assigning the appropriate value,
depending of it being multiple choice or not. If we were not able to evaluate
a certain criterion, we used the # symbol in the corresponding line in the
evaluation file. Currently, we only assign one value to criteria that might
accept multiple values, e.g. the data format criteria.

We obtained a total of 2,270 evaluations for all criteria over all tools, in
contrast to 2,883 possible criteria (93 tools x 31 criteria). This discrepancy
is contributed to our inability to evaluate all tools across all defined
criteria. This covers 78.7% of all possible evaluations.

Regarding the evaluation of each criteria, the coverage ranges from 38 for the
most exclusive criteria to 93 covered tools for most inclusive criteria. Five
criteria were evaluated for all tools, namely “available”, “online_available”,
“last_publication”, “citations_corpora” and “citations”. The criteria that we
most missed were “workable” (38 evaluations), “multilingual” (38 evaluations),
and “installation” (41 evaluations). Additionally, some of there criteria
require hands-on experiments with the tools, which was not possible for all
tools.

We could evaluate all 31 criteria for 15 tools, as already reported in Neves
and Ševa (2019), and the average number of evaluated criteria was 24. RAD was
the tool for which we assessed the fewest number of criteria (only 8). Further
tools with less than 20 evaluated criteria were: Analec (13), AWOCATo (14),
eHost (15), Hexatomic (16), SFA (17), OLLIE (17), Coco (18), Cas Editor (18),
Slate (18), KAFnotator (18), and Serengueti (19).

###  2.3 Directory maintenance

We designed the directory focusing on the easiness of its maintenance. For any
modification, a user should raise an issue in the GitHub repository. Here we
discuss the possible changes that we can envisage for the repository.

#### Addition of new annotation tools.

New annotation tools can be added simply by placing its evaluation file, with
matching file name to the tools name, in the “tools” folder of the repository.
Requests for new additions can be performed by creating an issue in the GitHub
repository. Removing a tool is then, in contrast, simply a matter of removing
its evaluation file from the “tools” folder of the repository We cannot,
however, envisage a reason why a certain tool should be removed from the
repository. As already stated above, we currently include all tools that we
found. Although some tools are no longer available, we include them for the
sake of completeness.

#### Addition of new criteria.

A new criterion should first be inserted in the “schema” file and then
replicated for each tool’s evaluation file (“tools” folder). Adding the
additional criterion (line) for each tool’s file could be carried out either
manually or automatically via a script, for instance, by setting a default
value or leaving the corresponding criterion (line) commented out. However, a
missing criteria for a particular tool does not cause a problem for neither
the scripts nor the Web tool.

#### Changes to a criterion for a tool.

Due the number of tools and criteria, coupled with the manual nature of the
evaluation, we keep the possibility of a faulty evaluation open. In such
situations updating the evaluation is a matter of changing the appropriate
line(s) in the relevant evaluation file. We are glad to receive feedback from
the community regarding our evaluation.

##  3 Scripts and Web application

Figure 1: Screen-shot of the Web application and its filters (on the right, in
orange).

We implemented a Web application for searching among the tools. The system is
implemented two fold. We use Python/Flask for the server side logic which
consists of reading all evaluation files and serving them to the interface.
The interface was, in turn, implemented in AngularJS JavaScript framework as
was designed as a responsive, single-page web application. It supports
searching by all non-numerical criteria, omitting e.g. last publication year,
through a dynamic side navigation bar (cf. Figure 1). Initially all tools are
presented in a tabular way, with configurable pagination functionality. By
choosing at least one of the possible filters the list will be filtered to
tools matching selected criteria.

The matching tools are shown in the decreasing order of the number of search
criteria that were matched. For tools with the same number of matching
criteria, alphabetical ordering is used. For each tool, we print the number of
matched criteria and their name(s), as selected in the filtering options.
Additionally, the search can be adapted to use OR and AND truth-functional
operator against the selected criteria. Initially, OR is used. The Web
application is currently running in Heroku
333https://annotationsaurus.herokuapp.com/.

##  4 Use Cases

Figure 2: Screen-shot of the Web application for the use case 1. We show the
list of the top 10 tools returned when after selecting the filters that we
defined for the use case.

We describe two use cases to illustrate how the repository (and search tools)
could support finding the annotation tools that best fit one’s needs. For both
use cases, we require annotation tools to be Web-based and freely available
for research purposes (i.e. “type=web-based” and “available” in #requirements,
and “free” in #technical) and that allow the definition of a schema
(“schematic” in #requirements). Further, we experimented with two optional
criteria, namely, whether the tools are installable and workable (in
#requirements). We did not put any limitation for the availability of source
code or format for schema, documents or annotations.

#### Use Case 1: Semantic annotation of entity and relations

Our first scenario addresses a typical annotation project of semantic
annotations. This is frequently necessary for construction of domain-specific
corpora, e.g. Ohta et al. (2012); Boyce et al. (2012). This use case requires
tools for manual annotation of named-entities (i.e. highlighting of text
spans) and for drawing relations between these. Therefore, we included the
filters “highlight” and “relationships” (in #functional) as mandatory. Given
the complexity and ambiguity of many domains, we expect the tool to be able to
assign multiple labels to text spans (i.e. overlapping annotations), i.e., the
“multilabel” criterion (in #functional). Finally, such annotation projects are
demanding and might take some hours for a single (long) document, therefore,
we also require the tool to be able to save documents partially (“partial-
save” criterion in #functional).

Seven tools (brat, INCEpTION, Inforex, MAT, SANTO, TeamTat, and WebAnno)
matched the eight mandatory criteria that we defined, i.e. the four general
non-optional ones and the four specific ones. Further, when checking the two
general optional arguments that require the tools to be installable and
workable, we had only four tools matching all filters (brat, MAT, TeamTat, and
WebAnno).

#### Use Case 2: Document-level annotations for text classification

For our second use case, we envisage the annotation of clinical data, e.g.
pathology reports Jouhet et al. (2012) or death certificates Lavergne et al.
(2016). Thus, we require the criterion “data-privacy” (in #functional) that
assures that the tool can be locally installed. Further, we address the task
of annotation on the document level (criterion “document-level” in
#functional), such as carried out in Baker et al. (2016). As optional
additional criteria, we check whether these tools provide annotations based on
pre-defined ontologies, i.e. by specifying the criteria “preannotations” and
“ontologies” (in #functional).

Six tools (Bionotate, doccano, MAT, SMART, TALEN, and UniversalAnnotationTool)
complied with the six mandatory filters that we defined, i.e., the four
general ones and the two specific ones for the use case. When checking the
other two optional filters for this use case, no tools could comply with all
eight criteria, but 12 tools matched seven of these criteria. For instance,
SANTO and WebAnno did not comply with the document-level annotation, while
Bionotate and MAT do not support ontologies.

##  5 Conclusions

We described a directory of annotation tools with search functionality. It
currently includes a total of 93 tools that were evaluated on many of the 31
criteria. We developed a simple Web application to search among the tools by
filtering for chosen criteria. The tools’ evaluation files and the Web
application are available in a GitHub repository and we welcome feedback from
the community regarding missing tools, interesting new criteria, and
corrections of our current evaluations.

In spite of covering a total of 93 tools and 31 criteria, our repository have
many limitations. Our list of annotation tools is very comprehensive, but some
tools might be missing and the links to tools’ publication and Web sites might
become broken at any time. Although the evaluation (plain text) files are
readable by both human and scripts, this also makes the process error-prone if
the criteria name is not written exactly as defined in the schema file.
Therefore, we provide a script for checking errors in the criteria names, as
well as for printing statistics of the current state of the evaluation. For
the sake of simplicity, we decided for a file-based approach in this first
version of the directory. Later, if necessary, these files can be converted to
a more appropriate format or imported into a database.

We certainly missed some interesting criteria that could be useful for some
domains or annotation projects, such as support for annotation layers, or
tools suitable for annotation of messages from the social media when building
specific corpora O’Connor et al. (2014). Further, the definition of some
criteria could be improved, such as specifying the corpora which where
developed using a certain tool instead of simply informing how many were
found.

We also envisage future work for supporting the automatic update of some
criteria. For instance, the last version (year of publication or commit) of a
tool could be automatically extracted from the GitHub repository (if
available), the number of citations for a publication might be automatically
retrieved from Google Scholar, and the corpora which were developed using a
certain tool could potentially also be automatically extracted from
publications.

## References

  * Baker et al. (2016) Simon Baker, Ilona Silins, Yufan Guo, Imran Ali, Johan Högberg, Ulla Stenius, and Anna Korhonen. 2016.  Automatic semantic classification of scientific literature according to the hallmarks of cancer.  _Bioinformatics_ , 32(3):432–440.
  * Boyce et al. (2012) Richard Boyce, Gregory Gardner, and Henk Harkema. 2012.  Using natural language processing to extract drug-drug interaction information from package inserts.  In _BioNLP: Proceedings of the 2012 Workshop on Biomedical Natural Language Processing_ , pages 206–213. Association for Computational Linguistics.
  * Fort (2016) Karën Fort. 2016.  _Collaborative Annotation for Reliable Natural Language Processing: Technical and Sociological Aspects_ , 1st edition.  Wiley-IEEE Press.
  * Jouhet et al. (2012) Vianney Jouhet, Gautier Defossez, Anita. Burgun, Pierre Le Beux, P Levillain, Pierre Ingrand, and Vincent Claveau. 2012.  Automated classification of free-text pathology reports for registration of incident cases of cancer.  _Methods of information in medicine_ , 51 3:242–51.
  * Lavergne et al. (2016) Thomas Lavergne, Aurelie Neveol, Aude Robert, Cyril Grouin, Grégoire Rey, and Pierre Zweigenbaum. 2016.  A dataset for icd-10 coding of death certificates: Creation and usage.  In _Proceedings of the Fifth Workshop on Building and Evaluating Resources for Biomedical Text Mining (BioTxtM2016)_ , pages 60–69. The COLING 2016 Organizing Committee.
  * Neves and Leser (2012) Mariana Neves and Ulf Leser. 2012.  A survey on annotation tools for the biomedical literature.  _Briefings in Bioinformatics_ , 15(2):327–340.
  * Neves and Ševa (2019) Mariana Neves and Jurica Ševa. 2019.  An extensive review of tools for manual annotation of documents.  _Briefings in Bioinformatics_.  Bbz130.
  * O’Connor et al. (2014) Karen O’Connor, Pranoti Pimpalkhute, Azadeh Nikfarjam, Rachel Ginn, Karen L. Smith, and Graciela Gonzalez. 2014.  Pharmacovigilance on twitter? mining tweets for adverse drug reactions.  _AMIA … Annual Symposium proceedings / AMIA Symposium. AMIA Symposium_ , 2014:924–933.
  * Ohta et al. (2012) Tomoko Ohta, Sampo Pyysalo, Jun’ichi Tsujii, and Sophia Ananiadou. 2012.  Open-domain anatomical entity mention detection.  In _Proceedings of the Workshop on Detecting Structure in Scholarly Discourse_ , pages 27–36. Association for Computational Linguistics.
  * Yang and Li (2018) An Yang and Sujian Li. 2018.  SciDTB: Discourse dependency treebank for scientific abstracts.  In _Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 2: Short Papers)_ , pages 444–449. Association for Computational Linguistics.