Available via license: CC BY
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TYPE Original Research
PUBLISHED 21 March 2024
DOI 10.3389/fams.2024.1387012
OPEN ACCESS
EDITED BY
Kateryna Buryachenko,
Humboldt University of Berlin, Germany
REVIEWED BY
Ivan Kovalyov,
Osnabrück University, Germany
Evgeniy Petrov,
Institute of Applied Mathematics and
Mechanics (NAN Ukraine), Ukraine
*CORRESPONDENCE
Olesia Zavarzina
olesia.zavarzina@yahoo.com
RECEIVED 16 February 2024
ACCEPTED 04 March 2024
PUBLISHED 21 March 2024
CITATION
Langemann D and Zavarzina O (2024)
Expand-contract plasticity on the real line.
Front. Appl. Math. Stat. 10:1387012.
doi: 10.3389/fams.2024.1387012
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terms.
Expand-contract plasticity on the
real line
Dirk Langemann1and Olesia Zavarzina2*
1Institute of Partial Dierential Equations, Technische Universität Braunschweig, Braunschweig,
Germany, 2Department of Mathematics and Informatics, V.N. Karazin Kharkiv National University,
Kharkiv, Ukraine
The study deals with plastic and non-plastic sub-spaces Aof the real-line Rwith
the usual Euclidean metric d. It investigates non-expansive bijections, proves
properties of such maps, and demonstrates their relevance by hands of examples.
Finally, it is shown that the plasticity property of a sub-space Acontains at least
two complementary questions, a purely geometric and a topological one. Both
contribute essential aspects to the plasticity property and get more critical in
higher dimensions and more abstract metric spaces.
KEYWORDS
metric space, non-expansive map, plastic space, expand-contract plasticity, Banach
space
1 Introduction
Here, we investigate properties of plastic metric spaces. Shortly speaking, a metric space
is plastic if every non-expansive bijection is an isometry, cf. Section 2.
We will observe that the plasticity property consists of a geometrical sub-problem and
a topological sub-problem. That is the reason why plasticity of a metric space, which can
be easily defined, evolves as a challenging mathematical problem. In particular, we observe
that the plasticity of a metric space is not inherited from sup-spaces, i. e., from including
spaces, and it does not inherit to sub-spaces, i. e., to included spaces.
In this study, we concentrate on metric spaces which are sub-spaces of the real axis,
and in this apparently simple situation, the typical difficulties come to the light.
The probably first study devoted to the plasticity problem is the study mentioned in
the reference [1]; however, the term "plasticity" appeared much later and the problem itself
remained unnoticed for several decades. A short literature survey and the information
about the current progress in solution of the problem are shown in Section 2.2.
The study is organized as follows. Section 2 introduces the basic concepts and illustrates
the existence of non-expansive bijections in the case that the metric space is a union of
closed intervals. This case demonstrates the geometrical aspects of the problem. Then,
Section 3 discusses the plasticity of metric spaces by means of metric spaces which are
unbounded sequences of points, investigates the relevance of accumulation points and
continuous subsets, and attacks the more topological parts of the plasticity concept. Finally,
Section 4 resumes the observations and gives a short outlook to further research.
2 Basic concepts
We denote a metric space by (A,d) where Ais the set of points and d:A×A→
R+= {x∈R:x≥0}is the distance obeying the known axioms of positivity, symmetry,
non-degeneracy, and the triangle inequality.
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2.1 Non-expansive maps
A map ϕ:A→Afrom the metric space Ainto itself is called
non-expansive if
d(ϕ(x), ϕ(y)) ≤d(x,y) for all x,y∈A(1)
is fulfilled. If the equality holds for all pairs x,y∈A,ϕis an
isometry.
The condition in Equation (1) is equivalent to the Lipschitz-
continuity of the map ϕon Awith Lipschitz constant 1. Thus, a
non-expansive ϕis also continuous on A.
We will investigate metric spaces A⊆Aex which are embedded
in a metric sup-space (Aex,dex ) because the space Aex might be
known and well understood, and thus, its points or rather a
selection of them serve as elements of A. Now, it is obvious that
the restriction of the metric space (Aex,dex ) to the set Aleads to the
metric space (A,d) by the restriction of the distance d=dex|A×A
to the set A. It is less obvious whether a metric space (A,d) can be
extended to a sup-set Aex by choosing an appropriate dex. However,
it is always possible, to choose a function ˆ
dex :Aex ×Aex →R+,
which fulfills the properties of symmetry, non-degeneracy, and
positivity with ˆ
dex|A×A=d, which of course is not a metric in
general. Then, we can define the metric
e
dex(x,y)=inf
n,{z0,...,zn}"ˆ
dex(x,z0)+
n−1
X
i=0
ˆ
dex(zi,zi+1)+ˆ
dex(zn,y)#
as the infimum over all possible paths of arbitrary length between x
and y. However, such a metric e
dex may not really be an extension.
As in the real life, if one builds a new paths, which are shorter, the
old ones may no longer be used. In our notation, this means that it
may happene
dex(x,y)<d(x,y) for some x,y∈A.
Nevertheless, one may define a real extension dex of the metric
d, which is more artificial and a bit similar to the French railways
metric in the following way. Let us fix a point x0of the set Aand
define an arbitrary metric dAex on the set (Aex \A)∪ {x0}, which
might be the discrete metric or any other metric. Although less
intuitive, the needed extension is
dex =
d(x,y), for x,y∈A;
dAex (x,y), for x,y∈(Aex \A)∪ {x0};
d(x,x0)+dAex (x0,y) for x∈A,y∈(Aex \A).
existing and easily available. Therefore, we will not distinguish
between dex and din the following but use the distance din the
extended metric space and sub-space.
Oppositely, it is not evident whether the existence of a non-
expansive map ϕex :Aex →Aex provides a non-expansive map
ϕ:A→Abecause the simple restriction ϕ=ϕex|A, although
still Lipschitz continuous, is not necessarily a map into A. It might
happen that the image im ϕ=ϕ(A)⊆Aex is not a subset of A.
The opposite question whether a non-expansive ϕ:A→Acan
be extended to a non-expansive map on the extended space Aex
is the question about the extension of Lipschitz maps, preserving
the Lipschitz constant. In particular, it is always possible for real-
valued functions according to McShane’s extension theorem [2].
For functions from a subset of Rnto Rn, the extension to the whole
Euclidean space is possible due to Kirszbraun’s theorem [3]. We will
observe that non-expansive maps pose a lot of interesting questions
and some of them can be answered.
2.2 Plastic metric spaces
Let us define a plastic metric space.
Definition 2.1. A metric space A is called expand-contract plastic
(EC-plastic)—or just plastic—if every bijective non-expansive map
ϕ:A→Ais an isometry.
Definition 2.1 defines a plastic metric space Avia the non-
existence of any non-expansive bijection of the metric space Ato
itself, which is not an isometry. Some simple examples are the non-
plastic metric space A=Rwith the non-isometric non-expansive
bijective map ϕ:x7→ x/2 and the plastic metric space A=[0, 1] ⊂
Rwith exactly the two non-expansive bijections ϕ1=id. and
ϕ2:x7→ 1−x, which are both isometries.
The only general result concerning plasticity of metric space
states that every totally bounded metric space is plastic, see
Naimpally et al. [4] for details. In fact, in the study mentioned in
the reference [4], a more general result was obtained, i. e., so-called
strong plasticity of totally bounded metric spaces was shown.
Definition 2.2. A metric space Ais called strongly plastic if for
every mapping ϕ:A→Athe existence of points x,y∈Awith
d(ϕ(x), ϕ(y)) >d(x,y) implies the existence of two points ˜x,˜y∈A
for which d(ϕ(˜x), ϕ(˜y)) <d(˜x,˜y) holds true.
This property and its uniform version were researched in the
study mentioned in the reference [5]. It says that any expansion of
a distance between two points implies the existence of two other
points which are contracted by the map ϕ. Observe it is extremely
important not to interchange expansion and contraction.
In the study mentioned in the reference [6], the following
intriguing question was posed.
Problem 2.3. Is it true, that the unit ball of an arbitrary Banach
space is plastic?
Observe that in finite dimensions, this question is answered
positively since in finite dimensions, the unit ball is compact and
thus totally bounded. Moreover, the question is open only in the
infinite dimensional case and the following more general problem.
Problem 2.4. For which pairs (X,Y) of Banach spaces, every
bijective non-expansive map ϕ:BX(0) →BY(0) between the unit
balls is an isometry?
There are a number of relatively recent particular results,
devoted to these problems, see Angosto et al. [7], Haller et al. [8],
Kadets andd Zavarzina [9], Leo [10], and Zavarzina [11]. There
exists also a circle of problems connected with plasticity property
of the unit balls. In the study mentioned in the references [12] and
[13], the so called linear expand-contract plasticity of ellipsoids in
separable Hilbert spaces was studied, which means that only the
linear non-expansive bijections were considered in the definition
of plasticity.
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Many natural questions concerning plasticity seem to have no
answer or even have not yet been considered. In 2020, Behrends
[14] draw attention to the fact that nobody studied the subsets of
the real line with respect to the plasticity problem. He tried to attack
this problem and received some results in this direction, however,
decided not to publish them. Moreover, the following problem is
still open.
Problem 2.5. What characterizes plastic sub-spaces of the real line
Rwith the usual metric d?
In spite of the seeming simplicity of the question, it is not so
easy to deal with. Let us first list the previously known results. As
we mentioned before, the set Ritself with the usual metric is not
plastic. If one considers any bounded subset, it is already plastic
due to its total boundedness.
On the other hand, it is easy to show that the set of integers Z
with the same usual metric is plastic in spite of its unboundedness
and the set R\Z. The proof of the plasticity of both mentioned
spaces may be found in the study mentioned in the reference [4]. In
the proof of plasticity of the set R\Z, one of the possible cases was
missed; nevertheless, the statement is still correct.
Already, these examples show that there is no simple answer
to the question whether a metric space is plastic or not. Rather we
could give the interpretation that there are some critical points,
e. g., the integers in these examples, which every non-expansive
bijection ϕdefinitely has to pass, what relates to the geometry of
the metric space A, and that there are some parts of the metric
space which cannot be glued to each other such as singular points or
open intervals, what relates to the topological aspects of plasticity.
We observe that sub-spaces of the real axis are already sufficiently
multifaceted to study the plasticity problem of metric spaces.
The question whether more general metric spaces are plastic,
provoke analogous difficulties, and again contain geometrical and
topological aspects.
Here, we will generalize the known results and say something
more about plastic sub-spaces of the real line. The previously
mentioned results explain why we consider only unbounded sets
in what follows.
All over the text, we use the notion dfor the usual Euclidean
metric d(x,y)= |x−y|for x,y∈R. Round brackets denote open
intervals (x,y)= {z∈R:x<z<y}and square brackets denote
closed intervals [x,y]= {z∈R:x≤z≤y}.
2.3 A subset of the real axis
We have observed that the real axis Rhas sufficiently interesting
metric sub-spaces for the investigation of plasticity. The Lipschitz
condition in Equation (1) lets us easily decide whether a map
ϕ:R→Ris non-expansive or not—just by the graph of the map
ϕ, see Figure 1. Due to our considerations in Section 2.1, which is
applied here with Aas union of intervals and Aex =R, the map
ϕcan be extended—not necessarily in a unique manner—as non-
expansive function ϕex on the entire axis R. Thus, ϕex is continuous
on R.
Figure 1 shows examples of bijective maps from the union of
intervals A=... ∪[a2,a3]∪[a4,a5]∪... ⊂Ronto itself. In
this example, the closed interval and the interspaces have increasing
lengths, in detail aℓ+1−aℓ≥aℓ−1−aℓ−2for all ℓ∈Z. Due to its
continuity, every bijection ϕpasses monotonically a rectangle in
A×A. In this example, with increasing lengths of the respective
intervals, we easily detect particular extensions ϕex :R→Rwith
ϕex|A=ϕand a slope bounded by 1 because the endpoints of
the interspace could be used in Equation (1). Hence, the functions
id. and ϕi,i=1, 2 below the diagonal are non-expansive, and the
function χabove the diagonal is expansive.
3 Main results
Let us start with some interesting observations on simple
situations of A, e. g., some sets of singular points.
Proposition 3.1. Let A= {ai}+∞
i=−∞ ⊂Rbe an increasing sequence
that obeys
d(ai−1,ai)≤d(ai,ai+1) for all i∈Z(2)
and
d(aj−1,aj)<d(aj,aj+1) for at least one i∈Z. (3)
Then (A,d) is not plastic.
Proof. The shift ϕ:ai7→ ai−1is an example of a non-expansive
bijection which is not an isometry.
Remark 3.2. The relation sign in Equations (2),(3) might be
commonly inverted so that the distances between two subsequent
points of Adecrease instead of increase, and the statement remains
unchanged.
Furthermore, let us consider sets which are bounded from one
side. Let us recall the definition of an accumulation point, which we
will use in what follows.
Definition 3.3. An accumulation point (or limit point) of a set A
in a metric space Xis a point x, such that every neighborhood of x
with respect to the metric on Xcontains a point of Awhich differs
from the point x.
An accumulation point of a set Adoes not have to be an element of
A. We will proceed with the following lemma.
Lemma 3.4. Let A⊂Rbe a set without accumulation points
which is bounded from one side. Let abe a minimal—or maximal—
element of Aand ϕ:A→Abe a bijective non-expansive map.
Then ϕ(a)=a.
Proof. Without loss of generality, we may consider the case when a
is a minimal element. Assume ϕ(a)6= a. Then there is b∈Asuch
that ϕ(b)=a.
Claim: Let be c ∈A. Then c ≤b implies ϕn(c)≤b for every
n∈N.
Proof of the Claim: We will use the induction in n. Indeed, if
ϕn(c)≤band ϕn+1(c)>bwe have
d(ϕn(c), b)≥d(ϕn+1(c), a)>d(b,a).
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FIGURE 1
Non-expansive maps ϕ1,ϕ2, and id. and an expansive map χfor a union A⊂Rof closed intervals of increasing length. The Cartesian product A×Ais
given in gray, and the bijections are black.
This contradiction completes the proof of the Claim.
Since
d(a,b)≥d(ϕ(a), ϕ(b)) =d(ϕ(a), a),
we have ϕ(a)≤b. Thus, the Claim provides ϕn(a)≤bfor every
n∈N. Now, the segment [a,b] is a trap for those points, which
were mapped there. Our aim is to find such a “trapped” point out
of the interior of the segment [a,b] and show that this leads to a
contradiction. There are only two possible cases.
Case 1: ϕ(a)=b. In this case, points aand bwere swapped
by ϕ. Then, such a “trapped” point is the closest from the right-
hand side point to b. There is c>bsuch that d(b,c)<d(b,d)
for any d>b. Such point cexists since Ais unbounded from
above and there is no accumulation points. The point ccannot be
mapped outside the segment [a,b] since it gives the contradiction
with non-expansiveness of ϕ.
Case 2: ϕ(a)<b. With such a condition, a “trapped” point is
ϕ(a) itself.
In both cases, we have a point twhich does not belong to the
interior of the segment [a,b] such that ϕ(t) belongs to this interior.
Consider an orbit of this point t, i. e., the set {ϕn(t)}∞
n=1. Due to the
bijectivity of ϕ, this orbit does not have repeating elements. Thus,
we have obtained a bounded infinite subset in Awhich contradicts
the fact that Adoes not have accumulation points.
Remark 3.5. The condition about the absence of accumulation
points in Lemma 3.4 cannot be omitted.
This remark is confirmed by the following example.
Example 3.6. Let A=Z+∪Q, where Q= { 1
4+1
n,n≥4}. The
bijective non-expansive map ϕis
ϕ(a)=
a−1, for a∈N,
1
2, for a=0,
1
4+1
n+1, for a=1
4+1
n∈Q.
We observe that ϕis bijective and it does not save the minimal
element of A. We check that it is non-expansive.
1. For all a,b∈N, the isometry d(ϕ(a), ϕ(b)) =d(a,b) is valid.
2. For a∈N,b=0, it holds d(ϕ(a), ϕ(b)) = |a−3
2|<a=d(a,b).
3. For a∈N,b=1
4+1
n∈Q, we have d(ϕ(a), ϕ(b)) =
|a−5
4−1
n+1|<|a−1
4−1
n| = d(a,b).
4. For a=0, b=1
4+1
n∈Q, it holds d(ϕ(a), ϕ(b)) = | 1
4−1
n+1|<
|1
4+1
n| = d(a,b).
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5. In the case a=1
4+1
n∈Q,b=1
4+1
m∈Q, without loss of
generality we may assume n<m. Then
d(ϕ(a), ϕ(b)) =1
n+1−1
m+1<1
n−1
m=d(a,b).
The described set is shown on the left of Figure 2.
Lemma 3.4 immediately implies the following corollary.
Corollary 3.7. Let A⊂Rbe an unbounded set without
accumulation points. Let Ahave a minimal or maximal element
and let ϕ:A→Abe a bijective non-expansive map. Then, ϕis
an isometry, moreover, the identity.
Proof. Without loss of generality, we may consider the case when
ais a minimal element. Let us show that ϕ(x)=xfor every
x∈A. Indeed, for the minimal element a, Lemma 3.4 ensures
that ϕ(a)=a. Now suppose for some fixed y∈A, the condition
ϕ(x)=xholds for every x<y,x∈A. Consider
A1=A\n[
x∈A,x<y
{x}o.
Then, ϕ|A1:A1→A1is a bijective non-expansive map, and y
is a minimal element. Then ϕ(y)=ydue to Lemma 3.4.
Proposition 4.1 in Naimpally et al. [4] states that for convex (in the
sense of the same study) metric spaces, hereditarily EC-plasticity
implies boundedness. Moreover, for convex subsets in Euclidean
Rn, hereditarily EC-plasticity and boundedness are equivalent.
However, the authors note that convexity is a too strong condition.
In Naimpally et al. [4], Theorem 4.3 states that an unbounded
metric space with at least one accumulation point contains a non-
plastic subspace. Corollary 3.7 demonstrates that the presence of an
accumulation point is essential in the mentioned theorem, since it
allows to build examples of unbounded hereditarily plastic spaces.
Let us go back to Example 3.6 and remark another interesting
property of non-expansive bijections on R. Suppose we have a set
A⊂Rand a function ϕ:A→A. We will say that ϕpreserves the
relation “between” on the set Aif for any x,y,z∈Awith x<y<z
we have ϕ(x)< ϕ(y)< ϕ(z). Example 3.6 shows that non-
expansive bijections do not have to preserve the relation “between.”
Surprisingly, there is an example demonstrating the same property
with a set without any accumulation points.
Example 3.8. Let A=N∪Q, where Q= {2k,k∈Z−}. The
bijective non-expansive map ϕis defined by
ϕ(a)=(a+6, if a≤ −4,
a+3, otherwise.
The map ϕdoes not preserve the relation “between” since
−4<−2<0 but ϕ(−2) < ϕ(−4) < ϕ(0). Let us check that ϕ
is non-expansive.
1. If both a,b≥ −2 or both a,b≤ −4, the non-expansiveness of ϕ
is obvious.
2. If a≥ −2 and b≤ −4, d(ϕ(a), ϕ(b)) = |a−b−3| ≤ |a−b| =
d(a,b). Only for a= −2 and b= −4, the inequality a−b<3
is valid, but even in this case, the previous inequality is true.
The described set is shown on the right of Figure 2.
Furthermore, we are going to present a sufficient condition for
a set in Rto be plastic. Let us introduce the set
DA= {p∈R:p=d(a,b) for some a,b∈Awith [a,b]∩A= {a,b}}.
Obviously, several pairs of points may be situated in the same
distance. That is why for every p∈DA, we call its multiplicity the
number of pairs of points in Awhich are on the distance p. This
multiplicity may be finite or infinite.
Theorem 3.9. Let A⊂Rhas no accumulation points and let DA
has a maximal element of finite multiplicity or a minimal element
of finite multiplicity. Then, (A,d) is a plastic metric space.
Proof. Without loss of generality, we may assume that DAhas a
minimal element a∈Rof finite multiplicity k∈N. Let us denote
Xa= {xn∈A,n=1, ..., 2k,d(xi,xi+1)=a,i=1, 3, ..., 2k−1}.
Let us take xi≤xjfor all i,jwith 1 ≤i<j≤2k. Consider
an arbitrary non-expansive bijection ϕ:A→A. Due to the non-
expansiveness of ϕ, we may conclude that ϕmaps Xabijectively
onto itself. Thus, ϕ|Xais an isometry on Xa. In particular, we find
d(x1,x2k)=d(ϕ(x1), ϕ(x2k)). Since this distance is the biggest one
on Xa, either ϕ(x1)=x1and ϕ(x2k)=x2kor ϕ(x1)=x2kand
ϕ(x2k)=x1. We will refer them as cases 1 and 2, respectively. In
the first case, obviously, for every x∈Awith x1<x<x2k, we get
ϕ(x)=x, so, in this case, ϕ|[x1,x2k]∩Ais the identity. In the second
case, if the structure of Aallows it, ϕ|[x1,x2k]∩Ais the inversion, called
total symmetry. Furthermore, following the similar procedure as in
Lemma 3.4, we have that in the first case, ϕis the identity, and in
the second case ϕis the total symmetry.
Remark 3.10. The conditions of Theorem 3.9 are sufficient but not
necessary for the plasticity of a set without accumulation points.
To make sure that the previous Remark 3.10 is true, one may
consider the space (Z,d). For DZ, the minimal and the maximal
elements are equal to 1 and have infinite multiplicity, but the space
is plastic. However, we constructed the next example, which is less
trivial, to show that plastic spaces which do not satisfy the condition
of the previous theorem may have richer structure.
Example 3.11. Let A= {ai}i=∞
i=−∞ ⊂R, where {ai}i=∞
i=−∞ is an
increasing sequence such that
d(ai,ai+1)=
|k| + 1, for i=2k,k∈Z,
1
k+1, for i=2k−1, k∈N,
1
|k|+2, for i=2k−1, k∈Z−.
The corresponding DAhas no minimal or maximal element.
However, (A,d) is plastic. In fact, let ϕ:A→Abe a non-expansive
bijection. Then,
d(ϕ(a0), ϕ(a1)) ≤d(a0,a1)=1.
Suppose d(ϕ(a0), ϕ(a1)) =1
n, where n≥2. Consider the open
ball with the radius n−1 centered in ϕ(a0). Due to the structure of
A, this ball contains only the point ϕ(a1), except for the center. On
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FIGURE 2
(Left) Illustration of Example 3.6. (Right) Illustration of Example 3.8. The gray dots on the axes indicate A. The black dots mark the respective
bijection. Clearly, no connection of two points has a slope larger than 1.
the other hand, the open ball with the radius n−1 centered in a0,
and for n≥3, it contains more than two points, and for n=2, it
contains two points but does not contain a1. In both cases, we have
a contradiction to the non-expansiveness of the map ϕ. That is why
the only possible option is as follows:
d(ϕ(a0), ϕ(a1)) =d(a0,a1)=1.
Furthermore, just in the same way as in Theorem 3.9, we have
that ϕis either the identity or the inversion.
Now let us speak about the subsets which contain a continuous
part. One may prove the following statement in the same way as
the Proposition 3.1.
Proposition 3.12. Let be
A=
+∞
[
i=−∞
(ai,bi)⊂R,
where bi<ai+1be such a sequence of intervals that
d(ai,bi)≤d(ai+1,bi+1) (4)
and
d(bi−1,ai)≤d(bi,ai+1) (5)
for all i∈Z. Furthermore, there exists j∈Zsuch that
d(aj,bj)<d(aj+1,bj+1) or d(bi−1,ai)<d(bi,ai+1). (6)
Then, (A,d) is not plastic.
Remark 3.13. In the same way as in Proposition 3.1, the relation
signs in Equations (4-6) might be commonly inverted.
Here is one more observation.
Proposition 3.14. Let A⊂Rcontain an interval (a,+∞) or
(−∞,a). Then, (A,d) is not plastic.
Proof. Without loss of generality, we discuss the case with (a,+∞).
Let us define the map ϕwith
ϕ(x)=(ϕ(x)=x, if x/∈(a,+∞),
ϕ(x)=x+a
2, otherwise.
This map is non-expansive, bijective, and, at the same time, not
an isometry.
In Naimpally et al. [4], Theorem 3.9 shows the plasticity of the
space R\Z. Unfortunately, the proof misses the case that the
non-expansive bijection is a symmetry. However, the statement
itself is true. One may use the same reasoning to prove the next
proposition.
Proposition 3.15. Let
A=
+∞
[
i=−∞
(ai,bi)⊂R,
where
d(ai,bi)=d(ai+1,bi+1) and d(bi,ai+1)=d(bi−1,ai).
Then, (A,d) is plastic.
Remark 3.16. Propositions 3.12 and 3.15 hold true with the closed
intervals.
Remark 3.17. On the other hand, if we consider in the statement of
Proposition 3.15 half-intervals,
A=
+∞
[
i=−∞
[ai,bi)⊂Ror A=
+∞
[
i=−∞
(ai,bi]⊂R
(A, d) is already a non-plastic space.
Frontiers in Applied Mathematics and Statistics 06 frontiersin.org
Langemann and Zavarzina 10.3389/fams.2024.1387012
FIGURE 3
Oppositely to Figure 1, half-open intervals allow that ϕdoes not pass entire rectangles in A×A. Rather, it might jump where the intervals can be
glued to each other. Remark that this example contains a first half-open interval and all the following intervals are half-open, cf. bijectivity. The
topological properties of the intervals in Aenter the plasticity problem.
Remark 3.18. If we consider in the same statement the set of the
form,
A=
n
[
i=−∞
[ai,bi]∪
+∞
[
i=n+1
(ai,bi]⊂R, where n∈N,
(A, d) is also a non-plastic space.
Figure 3 illustrates the previous remark.
The reader easily provides more examples which consist of
open or closed intervals together with half-intervals, all with the
same lengths. Again, we remark that the end-points of the intervals
are critical points for the plasticity property.
4 Conclusion
The analysis of plastic sub-spaces Aof the real-line Rhas shown
that first, the Lipschitz continuity of the map ϕ:A→Awith
Lipschitz constant 1 leads to useful and instructive illustrations of
the non-expansivity of the map ϕ, to which it is identical.
The plasticity property of a metric space turned out
to contain two complementary aspects, a purely geometrical
one and a topological one. Already on the real-line R, the
different nature of both aspects become visible. Whereas the
geometrical aspect is an extension of the non-expansivity of
ϕon a simply connected interval, the topological aspect leads
to the question whether two or more sub-intervals can be
glued at critical points by piecewise translations. Therefore, the
investigation of sub-spaces of the real-line Rgives an appropriate
framework for the investigation of the plasticity of metric
spaces.
We expect that the interplay between the two types of nature of
the problem gets more severe in higher dimensions. Already unions
of rectangles and cuboids as sub-spaces of the d-dimensional
Euclidean space Rdgive a tremendous multiplicity of open, half-
open, and closed edges and sides—complete or partial.
The named interplay between geometry and topology of the
metric spaces gets more and more complicated and less intuitive the
more abstract and the more elaborated the metric spaces are. We do
not expect any clarification, for example, metric spaces of functions
before sub-spaces of the Euclidean spaces are understood.
Future research will concentrate on the question, what else can
be said about plastic and non-plastic sub-spaces of the space (R,d).
Furthermore, we will explore the extension of a metric space Ato
larger sets in Aex which contain A. In particular, the metric hull, i. e.,
Frontiers in Applied Mathematics and Statistics 07 frontiersin.org
Langemann and Zavarzina 10.3389/fams.2024.1387012
the set
hullAex (A)=x∈Aex :∃y,z∈A:d(y,z)=d(y,x)+d(x,z)
⊆Aex,
gives interesting perspectives in the context of the plasticity
problem for the specification Aex =R. We conjecture that the
metric hull is the smallest proper extension of the metric space,
which is simply connected to Aex and where the plasticity is
dominated by the geometry. Therefore, the topology might be sub-
ordinated. In the medium term, we hope for an insight into the
question how geometry and topology interact in the plasticity of
a metric space.
Data availability statement
The original contributions presented in the study are included
in the article/supplementary material, further inquiries can be
directed to the corresponding author.
Author contributions
OZ: Writing – review & editing, Writing – original draft,
Project administration, Formal analysis, Conceptualization. DL:
Validation, Writing – review & editing, Writing – original draft,
Visualization, Methodology.
Funding
The author(s) declare financial support was received for the
research, authorship, and/or publication of this article. The research
was partially supported by the Volkswagen Foundation grant
within the frameworks of the international project “From Modeling
and Analysis to Approximation.” OZ was also partially supported
by Akhiezer Foundation grant, 2023.
Acknowledgments
The authors are grateful to Vladimir Kadets for valuable
remarks and pointing us the results about the extension of Lipschitz
maps. The authors are also thankful to Ehrhard Behrends for
drawing our attention to the problem considered in this study.
Conflict of interest
The authors declare that the research was conducted
in the absence of any commercial or financial relationships
that could be construed as a potential conflict of
interest.
Publisher’s note
All claims expressed in this article are solely those
of the authors and do not necessarily represent those of
their affiliated organizations, or those of the publisher,
the editors and the reviewers. Any product that may be
evaluated in this article, or claim that may be made by
its manufacturer, is not guaranteed or endorsed by the
publisher.
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