34

Dirac masses and isometric rigidity
György Pál Gehér

Tamás Titkos

Dániel Virosztek

This work was supported by the Research Institute for Mathematical Sciences, a Joint Usage/ResearchCenter located in Kyoto University.

Introduction
The aim of this short note is to expound one particular issue that was discussed during the talk [10]given at the symposium “Researches on isometries as preserver problems and related topics” at Kyoto
RIMS. That is, the role of Dirac masses by describing the isometry group of various metric spaces ofprobability measures. This article is of survey character, and it does not contain any essentially new
results.

From an isometric point of view, in some cases, metric spaces of measures are similar to C(K) ‐typefunction spaces. Similarity means here that their isometries are driven by some nice transformationsof the underlying space. Of course, it depends on the particular choice of the metric how nice thesetransformations should be. Sometimes, as we will see, being a homeomorphism is enough to generate an
isometry. But sometimes we need more: the transformation must preserve the underlying distance as
well. Statements claiming that isometries in questions are necessarily induced by homeomorphisms arecalled Banach‐Stone‐type results, while results asserting that the underlying transformation is necessarily
an isometry are termed as isometric rigidity results.
As Dirac masses can be considered as building bricks of the set of all Borel measures, a natural
question arises: Is it enough to understand how an isometry acts on the set of Dirac masses /? Does thisaction extend uniquely to all measures!? In what follows, we will thoroughly investigate this question.

1

Notions, notations
In this section we introduce all the notions and notations that are necessary to read the paper. Let

X\neq\emptyset be a set, and let \rho : X^{2}arrow \mathbb{R}_{+} be a metric on X . In our considerations, the metric topology on X
will always be complete and separable, so in order to simplify some notions, we assume that (X, \rho) is a
Polish space. The symbols \mathcal{P}(X) and \mathcal{M}(X) stand for the sets of probability measures and nonnegative
finite measures on the Borel a‐algebra of X , respectively. Given a measure \mu , the support S_{\mu} is the setof all points x\in X for which every open neighbourhood of x has positive measure.
As usual, \delta_{x} denotes the Dirac measure concentrated to x\in X . The set of all Dirac measures will bedenoted by \triangle(X) .
If a metric space (Y, d) is given, a map f : Yarrow Y is called an isometric embedding if it preserves the
distance, that is, d(f(x), f(y))=d(x, y) for all x, y\in Y. Surjective isometric embeddings are termed as
isometries.

For a measurable map \psi : Xarrow X , the push‐forward \psi_{\#} : \mathcal{P}(X)arrow \mathcal{P}(X) is defined by (\psi_{\#}(\mu))(A)=\mu(\psi^{-1}[A]) , where A\subseteq X is a Borel set, and \psi^{-1}[A]=\{x\in X|\psi(x)\in A\} . We call a metric space ofmeasures isometrically rigid, if all their isometries are of the form \psi_{\#} for some isometry \psi : Xarrow X.
A map f : \mathcal{P}(X)arrow \mathcal{P}(X) is called shape preserving if for all \mu\in \mathcal{P}(X) there exists a \psi\in Isom(X)(depending on \mu ) such that f(\mu)=\psi_{\#}(\mu) . An isometry is called exotic if it is not shape preserving.

35
The cumulative distribution function and its right‐continuous generalized inverse are key notions of
this short note. We recall these well known notions in the following two special cases: when X=\mathbb{R} andwhen X=[0,1] . If (X, \varrho)=(\mathbb{R}, | |) , the cumulative distribution function of \mu\in \mathcal{P}(\mathbb{R}) is defined as

F_{\mu}(x):=\mu((-\infty, x]) (x\in \mathbb{R}).

Its right‐continuous generalized inverse is defined as F_{\mu}^{-1}(y) := \sup\{x\in \mathbb{R} : F_{\mu}(x)\leq y\} for y\in(0,1) . If
(X, \varrho)=([0,1], | |) , we consider F_{\mu} and F_{\mu}^{-1} as [0,1]arrow[0,1] functions. In this case, F_{\mu}^{-1} is defined byright‐continuity at 0 and it takes the value 1 at 1.

2

Banach-Stone ‐type theorems and isometric rigidity

In this section we will provide some examples of Banach‐Stone‐type and isometric rigidity results fromthe last decade. We do not wish to give a complete overview of the recent progress in this flourishing
field, we consider only those results which are closely related to our organizing principle. Namely, therole of Dirac masses.

We start by highlighting an idea of Molnár, which is some kind of core of the results listed in thissection. Assume that we have a metric d on \mathcal{P}(\mathbb{R}) , and consider a set S\subset Y . Defineu(S) := { \nu\in \mathcal{P}(X)|d(\nu, \mu)=1 for all \mu\in S },and observe that, if \phi is a distance preserving bijection on \mathcal{P}(\mathbb{R}) with respect to d , then the cardinality ofu(u(\{\mu\})) and u(u(\{\phi(\mu)\})) are the same. Consequently, the following characterization (which is valid
for the Kolmogorov‐Smirnov, Kuiper, and Lévy metrics) guarantees that an isometry restricted to \triangle(X)is a bijection of \triangle(X) :\mu\in\triangle(X) \Leftrightarrow u(u(\{\mu\}))=\{\mu\}.
After this important remark we proceed with two Banach‐Stone type theorems.
Kolmogorov‐Smirnov distance d_{KS} on \mathcal{P}(\mathbb{R}) is defined by

We recall that thed_{KS}( \mu, \nu):=\Vert F_{\mu}-F_{\nu}\Vert_{\infty}=\sup_{x\in \mathbb{R} |F_{\mu}(x)-F_{\nu}(x)|.
The following characterization was obtained by Dolinar and Molnár in [2].
Theorem 1. Let \phi : \mathcal{P}(\mathbb{R})ar ow \mathcal{P}(\mathbb{R}) be a Kolmogorov‐Smirnov isometry, that is, a bijection on \mathcal{P}(\mathbb{R}) withthe property thatd_{KS}(\phi(\mu), \phi(\nu))=d_{KS}(\mu, \nu) (\mu, \nu\in \mathcal{P} (\mathbb{R})).

Then either there exists a strictly increasing bijection \psi : \mathbb{R}arrow \mathbb{R} such that

F_{\phi(\mu)}(t)=F_{\mu}(\psi(t)) (t\in \mathbb{R}, \mu\in \mathcal{P} (\mathbb{R})) ,(1)

or there exits a strictly decreasing bijection \tilde{\psi} : \mathbb{R}arrow \mathbb{R} such that

F_{\phi(\mu)}(t)=1-F_{\mu}(\tilde{\psi}(t)-) (t\in \mathbb{R}, \mu\in \mathcal{P}(\mathbb{R})) ,where F_{\eta}(x-) denotes the left limit of the distribution function F_{\eta} at the point
formation of the form (1) or (2) is a Kolmogorov‐Smirnov isometry.

(2)x. Moreover, any trans‐

A recent work concerning the closely related Kuiper metric provides a Banach‐Stone‐type result as
well. Recall that the Kuiper distance of \mu, \nu\in \mathcal{P}(\mathbb{R}) is given by the formulad_{K}(\mu, \nu):= \sup_{I\in \mathcal{I} |\mu(I)-\nu(I)| ,where\mathcal{I}= { I\subset R|\# I>1 and I is connected}.

Now, the characterization of Kuiper isometries reads as follows. (For more details see [3].)36
Theorem 2. Let \phi : \mathcal{P}(\mathbb{R})ar ow \mathcal{P}(\mathbb{R}) be a Kuiper isometry, that is, a bijection on \mathcal{P}(\mathbb{R}) with the property
thatd_{K}(\phi(\mu), \phi(\mu))=d_{K}(\mu, \nu) (\mu, \nu\in \mathcal{P}(\mathbb{R} )).

Then there exists a homeomorphism g:\mathbb{R}arrow \mathbb{R} such that\phi(\mu)=g_{\#}(\mu) (\mu\in \mathcal{P}(\mathbb{R})).

Moreover, every transformation of this form is a Kuiper isometry on \mathcal{P}(\mathbb{R}) .
Before continuing, let us make an observation. For any two real numbers x\neq y we have d_{KS}(\delta_{x}, \delta_{y})=1
and d_{K}(\delta_{x}, \delta_{y})=1 regardless to the value of |x-y| . This means that although \mathcal{P}(\mathbb{R}) does contain anatural copy of \mathbb{R} , the embedding x\mapsto\delta_{x} does not need to carry over any metric information from X.
As it will turn out soon, the form of isometries changes radically, once we consider a metric on \mathcal{P}(\mathbb{R}) thattakes care of distances attained in the underlying space.
The first rigidity result that we mention is about the Lévy distance. For \mu, \nu\in \mathcal{P}(\mathbb{R}) define

d_{L}( \mu, \nu) :=\inf\{\varepsilon>0|F_{\mu}(t-\varepsilon)-\varepsilon\leq F_{\nu}(t)\leq F_{\mu}(t+\varepsilon)+\varepsilon(\forall t\in \mathbb{R})\}.
Obviously, d_{L}( \delta_{x}, \delta_{y})=\min\{1, |x-y|\} , and thus it is not surprising at all that a Lévy isometry \phi :\mathcal{P}(\mathbb{R})ar ow \mathcal{P}(\mathbb{R}) must be related to an isometry of \mathbb{R} . In fact, Molnár proved that every Lévy isometry isimplemented by a translation and a reflection [9].
Theorem 3. Let \phi : \mathcal{P}(\mathbb{R})ar ow \mathcal{P}(\mathbb{R}) be a Lévy isometry, that is, a bijection on \mathcal{P}(\mathbb{R}) with the property
thatd_{L}(\phi(\mu), \phi(\nu))=d_{L}(\mu, \nu) (\mu, \nu\in \mathcal{P}(\mathbb{R} ))Then there is a constant c\in \mathbb{R} such that either

F_{\phi(\mu)}(t)=F_{\mu}(t+c) (t\in \mathbb{R}, \mu\in \mathcal{P}(\mathbb{R}))(3)

F_{\phi(\mu)}(t)=1-F_{\mu}((-t+c)-) (t\in \mathbb{R}, \mu\in \mathcal{P} (\mathbb{R}))

(4)

or

holds. Moreover, any transformation of the form (3) or (4) is a Lévy isometry on \mathcal{P}(\mathbb{R}) .
The second isometric rigidity result is about Borel probability measures living on real separable Banach
spaces endowed with the Lévy‐Prokhorov distanced_{LP}( \mu, \nu)=\inf { \varepsilon>0|\mu(A)\leq\nu(A^{\varepsilon})+\varepsilon for all A\in \mathcal{B}_{X} },whereA^{\varepsilon}= \bigcup_{x\in A}B_{\varepsilon}(x)and

B_{\varepsilon}(x)=\{y\in X|d(x, y)<\varepsilon\}.

Molnár’s trick on characterizing Dirac masses as measures satisfying u(u(\{\mu\}))=\{\mu\} works here as
well. Moreover, we have again that d_{LP}( \delta_{x}, \delta_{y})=\min\{1, |x-y|\} , but it is not so obvious for first sightthat an isometry acts like a distance preserving bijection on \triangle(X) . For the details see [4].

Theorem 4. Let (X, ||\cdot||) be a separable real Banach space and let \phi : \mathcal{P}(X)arrow \mathcal{P}(X) be a Lévy‐Prokhorov
isometry, that is, a bijection satisfyingd_{LP}(\phi(\mu), \phi(\nu))=d_{LP}(\mu, \nu) (\mu, \nu\in \mathcal{P}(X))
holds. Then there exists an affine isometry \psi : Xarrow X which induces \phi , that is, we have\phi(\mu)=\psi_{\#}(\mu) (\mu\in \mathcal{P}(X)) .
Moreover, any transformation of the form (5)i\mathcal{S}a Lévy‐Prokhorov isometry.

After these Banach‐Stone type and rigidity results one can have the feeling that‐ an isometry maps \triangle(X) onto \triangle(X)‐ the action on \triangle(X) determines the isometry uniquely
In what follows, we will see that none of these statements are true.

(5)

37
3

Quadratic Wasserstein spaces

First, we recall the notion of a Wasserstein space \mathcal{W}_{p}(X) . For a parameter value p\geq 1 let us denotethe set of Borel measures with finite pth moment by\mathcal{W}_{p}(X) :=\{\mu\in \mathcal{P}(X)|\exists x_{0}\in X : \int_{X}\rho (x, x_{0})^{p}d\mu(x)<\infty\},where (X, \rho) is a complete and separable metric space. A Borel probability measure \pi on X^{2} is a couplingfor \mu, \nu\in \mathcal{P}_{p}(X) (\pi\in C(\mu, \nu) , in symbols), if their marginals are \mu and \nu , i.e., for all Borel sets A\subseteq X
it satisfies\pi(A\cross X)=\mu(A)The setand\pi(X\cross A)=\nu(A) .

(6)d_{W_{p} ( \mu, \nu)=(\inf_{\pi\in C(\mu,\nu)}\int_{X^{2} \rho(x, y)^{p}d\pi(x, y) ^{1/p}(7)\mathcal{W}_{p}(X) endowed with the metricis called shortly as the p‐Wasserstein space (onX ).

One of the features of the metric d_{W_{p}} is that it

takes care of large distances in X . In fact, the embedding of X into \mathcal{W}_{p}(X) as the set of Dirac masses isdistance preserving. For more details and historical comments we refer the reader to [12].

From our point of view, the most important results were obtained by Bertrand and Kloeckner for

quadratic (p=2) Wasserstein spaces. In [7], Kloeckner provided a detailed study of quadratic Wassersteinspaces built on finite dimensional Euclidean spaces. According to his results, considering the quadratic
Wasserstein distance, none of the Wasserstein spaces built on Euclidean spaces are isometrically rigid.
Moreover, if the underlying Euclidean space is of dimension 1, then even exotic isometries exist. For moredetails see Section 5 in [7].
Theorem 5. The isometry group of the space \mathcal{W}_{2}(\mathbb{R}) is a semidirect product

Isom \mathbb{R}\ltimes Isom \mathbb{R} .

(8)

In (8) the left factor is the image of \# and the right factor consists of all isometries that fix pointwise the\mathcal{S}etof Dirac measures. Moreover, the right factor decomposes as Isom \mathbb{R}=C_{2}\ltimes \mathbb{R} , where the C_{2} factor(the group of order 2) is generated by a non‐trivial involution that preserve shapes and the \mathbb{R} factor is aflow of exotic isometries.

According to this desrciption, there are many isometries with identical action on Dirac masses, sothat it cannot be true that an isometry is determined by its action on Dirac masses.

Later, it turned out that negative curvature makes the structure of the isometries simpler [1] in thesense that the quadratic Wasserstein space built on a negatively curved geodesically complete Hadamard
space is isometrically rigid.

4

Splitting masses
After showing in [7] that \mathcal{W}_{2}(\mathbb{R}) admits exotic isometries, Kloeckner posed the following two questions.
Does there exist a Polish (or Hadamard) space X\neq \mathbb{R} such that \mathcal{W}_{2}(X) admits exotic isometries?
Does there exist a Polish space X whose Wasserstein space \mathcal{W}_{2}(X) possess an isometry that does
not preserve the set of Dirac masses?

38
In this short section we will highlight that the choice of parameter value p=2 is essential in thesequestions. In fact, we will prove by showing an example that the answer is affirmative for both questions
if

p=1.

Set X to be the unit interval [0,1] . The special feature of \mathcal{W}_{p}([0,1]) is that the p‐Wasserstein distanced_{W_{p}} can be calculated as

d_{W_{p} ( \mu, \nu)=(\int_{0}^{1}|F_{\mu}^{-1}(t)-F_{\nu}^{-1}(t)|^{p}dt) ^{\frac{ \imath} {p} (\mu, \nu\in \mathcal{W}_{p}([0,1]).

Furthermore, according to Vallender [11], in the special case of p=1 and X=[0,1] , the Wasserstein
distance can be calculated by means of the distribution functions as welld_{W_{1} ( \mu, \nu)=\int_{0}^{1}|F_{\mu}(t)-F_{\nu}(t)|dt (\mu, \nu\in \mathcal{W}_{1}([0,1]).

Recall that a cumulative distribution function of a \mu\in \mathcal{P}([0,1]) is monotone increasing, continuous fromthe right and takes the value 1 at the point 1. Conversely, any function F : [0,1]arrow[0,1] satisfying theabove three conditions is the cumulative distribution function of some Borel probability measure on [0,1].
Consequently, for any measure \mu\in \mathcal{P}([0,1]) , the function F_{\mu}^{-1} is a cumulative distribution function ofsome measure \nu\in \mathcal{P}([0,1]) , that is, F_{\nu}=F_{\mu}^{-1} It is easy to see that the map j : \mathcal{W}_{1}([0,1])arrow \mathcal{W}_{1}([0,1])
defined by the equation

F_{j(\mu)}=F_{\mu}^{-1} (\mu\in \mathcal{W}_{1}([0,1]))

preserves the distance. As joj is the identity of \mathcal{W}_{1}([0,1]) , we see also that j is a bijection, and thus anisometry.

\bullet F_{\mu}oF

Finally, observe that j does not send Dirac masses to Dirac masses. Indeed, (as it can be seen on the
figure), j(\delta_{t})=t\delta_{0}+(1-t)\delta_{1} for all 0\leq t\leq 1 . More details about isometries and isometric embeddingsof \mathcal{W}_{p}([0,1]) and \mathcal{W}_{p}(\mathbb{R}) spaces can be found in [6].

5Some remarks on isometric embeddings
We close this short note by mentioning our recent result on the discrete case [5]. Our aim to do so is

to show how difficult the description of distance preserving maps can be, when one drops bijectivity.
Let X\neq\emptyset be a countable set, and let \rho : X^{2}arrow\{0,1\} be the discrete metric, i.e., \rho(x, y) :=1 if x\neq yand \rho(x, x) :=0 for all x, y\in X . To avoid trivialities, we assume that X has at least two elements.
Before showing an example and stating the theorem, we emphasize that we do not assume affinity or
any other algebraic property when speaking about isometric embeddings.

39
Let us fix a parameter value p\in(0, \infty) , and let
discrete metric. Define

X be the set of natural numbers endowed with the

f:\mathcal{W}_{p}(X)arrow \mathcal{W}_{p}(X) asf ( \sum_{x\in S_{\mu}}c_{x} . \delta_{x})=\sum_{x\in S_{\mu}}[\ln(1+c_{x}) . \delta_{2x}+(c_{x}-\ln(1+c_{x}) . \delta_{2x+1}].
One can show by definition that this is a non‐surjective isometric embedding. (Observe that the range off does not contain Dirac masses.) What happens here is roughly speaking the following: f splits Diracmasses asf(\delta_{x})=\ln 2\cdot\delta_{2x}+(1-\ln 2)\cdot\delta_{2x+1},
and redistributes weights. On the one hand, if fx\neq y , then S_{f(\delta_{x})}\cap S_{f(\delta_{y})}=\emptyset , thus f induces a partitionof X , in fact, the support of f(\mu) is the disjoint union

S_{f(\mu)}= \bigcup_{x\in S_{\mu} S_{f(\delta_{x}) .
On the other hand, we see that if \mu(\{x\})=c_{x} , thenthenf(\mu)(\{2x, 2x+1\})=c_{x} , and if \mu(\{x\})\leq\nu(\{x\})f(\mu)|_{\{2x,2x+1\}}\leq f(\nu)|_{\{2x,2x+1\}}.
We will see that every non‐surjective isometric embedding looks like this in a particular sense. The actionof f on \triangle(X) will induce a partition and a family of nonnegative finite measures satisfying some special
properties. It can be seen easily that only the lack of surjectivity is responsible for such phenomena,
because bijective isometries are basically just permutations of the underlying space.

Theorem 6. Let p\in(0, \infty) be fixed, and let f : \mathcal{W}_{p}(X)arrow \mathcal{W}_{p}(X) be an isometric embedding, i. e.,

d_{W_{p}}(\mu, \nu)=d_{W_{p}}(f(\mu), f(\nu))for all\mu, \nu\in \mathcal{W}_{p}(X) .

(9)

Then there exists a unique family \Phi of measures indexed by the set X\cross(0,1 ], that is\Phi:=(\varphi_{x,t})_{x\in X,t\in(0,1]}\in \mathcal{M}(X)^{X\cross(0,1]}(10)

that satisfies the following properties(a) for all x\neq y:S_{\varphi_{x,1}}\cap S_{\varphi_{y,1}}=\emptyset(b) for all x\in X and t\in(0,1]:\varphi_{x,t}(X)=t(c) 0<s<t\leq 1 implies \varphi_{x,s}\leq\varphi_{x,t} for all x\in X,
and that generates f in the following \mathcal{S}ensef( \mu)=\sum_{x\in S_{\mu} \varphi_{x,\mu(\{x\})}for all\mu\in \mathcal{W}_{p}(X) .

(11)

Conversely, every X\cross(0,1 ]‐indexed family of measures satisfying properties (a) — (c) generates an i_{\mathcal{S} o ‐metric embedding via the formula (11).

40
Acknowledgements
This paper is part of a long term collaboration investigating the isometric structure of Wassersteinspaces. The authors would like to thank the warm hospitality and generosity of László Erdós and his
group at Institute of Science and Technology Austria.
T. Titkos wants to thank Oriental Business and Innovation Center ‐ OBIC for providing financial
support to participate in the symposium at the Kyoto RIMS.

Gy. P. Gehér was supported by the Leverhulme Trust Early Career Fellowship (ECF‐2018‐125),
and also by the Hungarian National Research, Development and Innovation Office (K115383). T.
Titkos was supported by the Hungarian National Research, Development and Innovation Office‐ NKFIH(PD128374), by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences, and by the
UNKP‐18‐4‐BGE‐3 New National Excellence Program of the Ministry of Human Capacities. D. Virosztek

was supported by the ISTFELLOW program of the Institute of Science and Technology Austria (projectcode IC1027FELL01 ) and partially supported by the Hungarian National Research, Development andInnovation Office NKFIH (grant no. K124152 and grant no. KH129601).
References[1] J. Bertrand, and B. Kloeckner, A geometric study of Wasserstein spaces: isometric rigidity in neg‐
ative curvature, Int. Math. Res. Notices 2016(5) (2016), 1368‐1386.
[2] G. Dolinar, and L. Molnár, Isometries of the space of distribution functions with respect to the
KolmogorovSmirnov metric, J. Math. Anal. Appl. 348 (2008), 494498.
[3] Gy. P. Gehér, Surjective Kuiper isometries, Houston Journal of Mathematics 44(1) (2018), 263‐281.
[4] Gy. P. Gehér, and T. Titkos, A characterisation of isometries with respect to the Lévy‐Prokhorov
metric_{Z} Annali della Scuola Normale Superiore di Pisa‐ Classe di Scienze (2018), in press.
[5] G. P. Gehér, T. Titkos, and D. Virosztek, On i_{\mathcal{S} ometric embeddings of Wasserstein spaces — thediscrete case, arXiv:1809.01101 (2018).
[6] G. P. Gehér, T. Titkos, and D. Virosztek, On isometric embeddings of Wasserstein spaces— the realline, Manuscript in preparation (2019).
[7] B. Kloeckner, A geometric study of Wasserstein \mathcal{S}paces : Euclidean spaces, Annali della Scuola Nor‐
male Superiore di Pisa‐ Classe di Scienze IX, 2 (2010), 297‐323.
[8] L. Molnár, Kolmogorov‐Smirnov isometries and affine automorphisms of spaces of distribution func‐
tions, Cent. Eur. J. Math. 9 (2011), 789‐796.
[9] L. Molnár, Lévy isometries of the space of probability distribution functions, J. Math. Anal. Appl.
380 (2011), 847‐852.
[10] T. Titkos, On the isometry group of metric spaces of probability measure\mathcal{S} , talk at the symposium:
Researches on isometries as preserver problems and related topics, Kyoto RIMS.

[11] S. S. Vallender, Calculation of the Wasserstein distance between probability distributions on the line,
Theory Probab. Appl. 18 (1973), 784786.
[12] C. Villani, Optimal Transport, Old and New, Springer, 2009.

[13] C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, vol. 58, American
Mathematical Society, Providence, RI, 2003.

[14] D. Virosztek, Maps on probability measures preserving certain distances — a survey and some new
results. Acta Sci. Math. (Szeged) 84 (2018), 65‐80.

41
41
György Pál Gehér
University of Reading, Department of Mathematics and Statistics
Whiteknights, P.O. Box 220,Reading RG66AX
United Kingdom
E‐mail address: G.P.Geher@reading.ac.uk or gehergyuri@gmail.com
Tamás Titkos

Alfréd Rényi Institute of Mathematics of the Hungarian Academy of Sciences
H‐1052 Budapest, Reáltanoda u. 13 ‐ 15andBBS University of Applied Sciences
H‐1054 Budapest, Alkotmány u. 9.
Hungary
E‐mail address: titkos.tamas@renyi.mta.hu
Dániel Virosztek

Institute of Science and Technology Austria
Am Campus 1, 3400 Klosterneuburg
Austria
Email address: daniel.virosztek@ist.ac.at