=Paper=
{{Paper
|id=Vol-1987/paper55
|storemode=property
|title=Spatial Equilibrium on the Plane and an Arbitrary Population Distribution
|pdfUrl=https://ceur-ws.org/Vol-1987/paper55.pdf
|volume=Vol-1987
|authors=Valeriy M. Marakulin
}}
==Spatial Equilibrium on the Plane and an Arbitrary Population Distribution==
https://ceur-ws.org/Vol-1987/paper55.pdf
Spatial Equilibrium on the Plane and an Arbitrary
Population Distribution
Valeriy M. Marakulin
Sobolev Institute of Mathematics, Russian Academy of Sciences
4 Acad. Koptyug avenue, 630090 Novosibirsk, Russia.
Novosibirsk State University,
2 Pirogova street, 630090 Novosibirsk, Russia.
marakul@math.nsc.ru
Abstract
The existence of immigration proof partition for communities (coun-
tries) in a multidimensional space is studied. This is a Tiebout type
equilibrium which existence previously was studied under weaker as-
sumptions (measurable density, fixed centers and so on). The migration
stability suggests that the inhabitants of frontier have no incentives to
change jurisdiction (an inhabitant at every frontier point has equal costs
for all possible adjoining jurisdictions). It is required that inter-country
border is represented by a continuous curve (surface). Assuming pop-
ulation is distributed in one or two dimension area (convex compact)
and this distribution is described by Radon’s measure, we prove that
for an arbitrary number of countries there exists stable partition into
countries. The proof is based on Kakutani’s fixed point theorem ap-
plied for specific approximation of initial problem with the subsequent
passing to the limits.
Introduction
In the seminal paper [Alesina & Spolaore, 1997] a basic model of country formation was offered. In this model,
the cost of the population is described as the sum of the two values—the ratio of total costs on the total weight
of the population plus transportation costs to the center of the state. This model has been studied in a number
of subsequent studies, but in each of them deals with the case of one-dimensional region and the interval-form
countries (country formation on the interval [0, 1]).
The first progress in the resolution of the problem of existence was obtained in [Le Breton et al., 2010], where
well known Gale–Nikaido–Debreu lemma was applied to state the existence of nontrivial immigration proof
partition for interval countries, i.e. such that no one has incentive to change their country of residence. In
[Le Breton et al., 2010] were made rather strong assumptions on the distribution of the population—continuous
density, separated from zero. Next in [Marakulin, 2017] mathematical part of the approach was significantly
strengthened and extended to the case of distribution of the population, described as a Radon measure (prob-
ability measure defined on the Borel σ-algebra). In [Savvateev et al., 2016] a new significant advancement was
Copyright ⃝
c by the paper’s authors. Copying permitted for private and academic purposes.
In: Yu. G. Evtushenko, M. Yu. Khachay, O. V. Khamisov, Yu. A. Kochetov, V.U. Malkova, M.A. Posypkin (eds.): Proceedings of
the OPTIMA-2017 Conference, Petrovac, Montenegro, 02-Oct-2017, published at http://ceur-ws.org
378
suggested, it disseminates the result (existence theorem) to the case of 2-dimensional (or more) region. The proof
of [Savvateev et al., 2016] is very elegant and is based on the application of KKM-lemma (Knaster–Kuratowski–
Mazurkiewicz), but the result is essentially limited by the presence of fixed in the space positions of capitals and
absolute continuous measure of the population distribution. The goal of our paper is to show how this result
can be extended to a more general case of Radon measure.
The problem of division of the rectangular area ABCD in R2 into two countries at a given measurable
random distribution of the population is studied. There is described a basic one-dimensional approximation, for
which a fixed point (via Kakutani’s theorem) can be found, and then the limit process gives the result. The
existence theorem and its proof for the population distribution, which is absolutely continuous with respect to the
Lebesgue measure, constitute the first section of the paper. The second section provides a further generalization
and extends the existence result to an arbitrary distribution of the population, described as a Radon measure.
The result generalises [Marakulin, 2017] to two-dimensional case. Indeed, the assumption for a measure to be
absolutely continuous with respect to the Lebesgue measure seems unrealistic one: What happens with the
urban population? There are no cities at all?.. and how the existence of unoccupied areas can hinder the
division into countries? It is clear that the reason for these misunderstandings has purely mathematical nature
and the solution can be found by applying an appropriate mathematical technique. This is the focus of this
work. Now the distribution of inhabitants is described using the Radon measure µ—this is a countably additive
probability measure defined on a Borel σ-algebra. By virtue of the theorem known in mathematical analysis,
the so-called Lebesgue decomposition, the measure µ can be represented as a sum of a purely discrete measure
ν and absolutely continuous with respect to the Lebesgue measure ϑ, i. e. µ = ν + ϑ. The absolutely continuous
component corresponds to the distribution of the rural population. At the same time, the purely discrete term ν
corresponds to the urban population, for then the measure has a counting carrier and is represented as the sum of
measures concentrated at a point (Dirac measure) of the form δ({a}) = α = δ(A) for all A⊆ABCD including
the point a, and zero otherwise. Here α can be understood as a population (mass) of the city, concentrated
(located) at the point a. Of course, from interesting considerations, we are only interested in the case of a
finite number of cities. And the last: what kind of divisions into countries exists and can be realized?—in
the formulation of [Le Breton et al., 2010], [Savvateev et al., 2016], this question does not matter because the
measure of single-point sets is zero. In our case this is not so, since there are points with non-zero mass. In this
case, the point-city can be divided by mass into two unequal parts, one of which belongs to one border state,
the other to another one. It can also happen a curious thing such as the emergence of country-cities of zero size
(area). Such cities as Jerusalem, Rome with the Vatican, previously—divided Berlin etc. can serve as a real
illustration of the theoretical conclusion.
1 The Partition into Two Countries on the Plane
The division of the one-dimensional world on countries can not be considered as a satisfactory solution of the
problem but two-dimensional formulation seems fundamentally more difficult. Now for a particular example of
division of the rectangular area in two countries we consider an approximating design allowing to find a solution
by passing to the limit.
First, we define the principle of stability applied for the country located on the plane. As in the case of one-
dimensional world, it must be such division that border residents have no incentive to change their jurisdiction.
Thus, the costs for any border resident should be the same with respect to any of the possible for her/him ad-
joining jurisdictions. It is assumed that the borders between the two countries allow continuous parametrization,
i.e. they are the image of the interval from R for some continuous one-to-one mapping. As a result, as in the
one-dimensional case, the function of individual costs of individuals should be continuous on the whole field of
country division, that is, country partition must implement continuous “gluing” of country-depended individual
costs.
For simplicity, we consider now a particular case of a rectangular area of possible settlement represented in
the diagram 1 rectangle ABCD. We assume that ci (x, y, ·), i = 1, . . . , n be the functions of individual costs,
depending on the place of individual location—defined via coordinates (x, y) ∈ ABCD, the weight of the
resident jurisdiction µi (Si ), the location of its center rc (Si ), metrics ρ(·, ·) (to specify the distance to the center),
cost of government gi and so on. The basic model representation of these cost functions is
gi
ci (x, y, µi (Si ), rc (Si )) = + ρ((x, y), rc (Si )), gi > 0, i ∈ N. (1)
µi (Si )
379
B q C
y1 x1
q
y2 x2
q
y3 x3
Sleft Sright
q
y4 x4
q
y5 x5
y6 x6q
A D
Figure 1: Possible division into two countries of the rectangular area ABCD, m = 6
In general these functions may have sufficiently general form but they always continuously depend on certain
country parameters and obey some other specific assumptions, see [Savvateev et al., 2016, Marakulin, 2016].
For the cost functions c1 (·), c2 (·) specified by (1) possible inter-countries boundaries are defined by equation
g2 g1
||(x, y) − rc (S1 )||2 − ||(x, y) − rc (S2 )||2 = − = const. (2)
µ(S2 ) µ(S1 )
Now a possible border has hyperbolic form and, for Euclidean distance, (2) defines the classic hyperbola. In the
next subsection we shall assume
(P) The distribution of population is described by an absolutely continuous probability measure µ such that
supp(µ) = ABCD.1
1.1 Partition of an Area on Plane via One-dimensional Approximation
The idea of approach is that given coordinate system (potentially curved), a stable partition relative to one-
dimensional world appeared along every coordinate line. At the same time, the function of individual costs
must be calculated relative to the position of “center” of the country and the general population distributed
in two-dimensional space. Finding such a partition is not an easy task, to solve this we shall apply a special
“one-dimensional approximation”, relative which a country partition can be found by a fixed point theorem
(Brouwer or Kakutani).
The construction as follows: specify m − 2 straight lines parallel to the base of the rectangle, m ≥ 3. Let
the lower base has a number m, top one—the number 1 and all others are numbered from top to bottom. Each
i-th segment is divided into two parts by the point xi , which can be considered the point from interval [0, 1]
(length of the base ABCD), i = 1, . . . , m. Straight line segments connecting consecutive points x1 , . . . , xm ,
form a polygon line, which we accept as the border between the left and right countries. Now, if density f (x, y)
is presented then it is possible to integrate it over each of the country area, finding the weights (size) µ(S) of
their populations.
Within each country its “center” (the capital) rc (S) ∈ S is specified, the position of which we will consider as
depending continuously from a given country settings x = (x1 , . . . , xm ) ∈ [0, 1]m . Thus we have:
∫
µ(Sleft ) = f (x, y)dxdy ≥ 0, rc (Sleft ) = rleft (x1 , . . . , xm ) ∈ Sleft
Sleft
∫
µ(Sright ) = f (x, y)dxdy ≥ 0, rc (Sright ) = rright (x1 , . . . , xm ) ∈ Sright .
Sright
Moreover, without loss of generality
µ(Sleft ) + µ(Sright ) = 1.
The fact that we are talking about the “mass of the population” (population) and the “distance to the center”
(transport availability of capital) as the main parameter determining the costs of individuals in a country, it is
only the interpretation of the cost function in the context of the main model variant. The same can be said
about the property of the center of the country be located on its territory — it’s just a natural variant of content,
∫
1 This combined means that µ(A) > 0 ⇐⇒ A dxdy > 0 for every measurable A⊆ABCD.
380
from a mathematical point of view, the center could be anywhere. What is really important is (described below)
certain properties of their individual costs.
Next, consider a point-to-set mapping, whose fixed point gives the desired country partition. The construction
of mapping applies the ideas borrowed from the one-dimensional case, see [Marakulin, 2017]. Let
X = [0, 1]m ,
and define a point-to-set mapping of X into itself.
Let c1 (·), c2 (·) be the functions of individual costs depending on the weight of the jurisdiction population
µ1 (x), µ2 (x), location of its center rc (S1 ), rc (S2 ), metrics ρ(·, ·) (to determine the distance to the center) and
a place of the individual location specified by coordinates (x, y) ∈ ABCD. The basic model representation of
these functions is (1). Now we shall think that they are functions of general form continuously depending on
x = (x1 , . . . , xm ) ∈ [0, 1]m for µ(Sk (x)) > 0, k = 1, 2. Additionally assume that
(i) ck (x, y, x) > 0 for µ(Sk ) ̸= 0 and
(ii) ck (x, y, x) → +∞ if µ(Sk ) → 0, k = 1, 2.
For the functions
∫ of (1) this condition
∫ is always satisfied. At the same time, if the density f (x, y) of the population
is so that A dxdy > 0 implies A f (x, y)dxdy > 0 for every measurable subset A ⊂ ABCD (i.e. each subset of
nonzero area (Lebesgue measure) has a population of non-zero mass), the latter requirement is equivalent to
c1 (x, y, x) → +∞ ⇐⇒ x → (0, . . . , 0) & c2 (x, y, x) → +∞ ⇐⇒ x → (1, . . . , 1). (3)
For the boundary points x1 , . . . , xm of country areas let us find an excess cost of possible (two) jurisdictions
(constants y1 , . . . , ym in the argument are excluded)
hi (x) = c1 (xi , x) − c2 (xi , x), i = 1, . . . , m.
Notice that (3) implies that for all i = 1, . . . , m, hi (x) → +∞ for x → 0, and x → 1 when hi (x) → −∞.
Next we define the (single-valued) map φ : X → X = [0, 1]m putting
{ hi (x)
xi − x2i · 1+h i (x)
, for hi (x) ≥ 0,
φi (x) = hi (x) (4)
xi + 2 · hi (x)−1 , for hi (x) ≤ 0.
1−xi
By construction, this mapping is well defined everywhere on X with the exception of two points x = 0 = (0, . . . , 0)
and x = 1 = (1, . . . , 1), which values can be specified by continuity:
φ(0) = (0, . . . , 0), φ(1) = (1, . . . , 1).
It is obvious that according to the construction these points are trivial fixed points of φ(·), that does not comply
with the requirements of the division of rectangular area. Further construction and analysis will focus on the
finding of the nontrivial fixed point corresponding to the division of the area into two countries with non-zero
masses of the population.
Now we define a point-to-set mapping Φ from X = X × ∆2 to X by formula: for (µ1 , µ2 ) =
(µ(Sleft (x)), µ(Sright (x))) specify
ν1
{ µ1 φ(x)}, if ν1 ≤ µ1 , µ1 ̸= 0,
Φ(x, ν) = { µν22 φ(x) + µ2µ−ν 2
(1, . . . , 1)}, if ν2 ≤ µ2 , µ2 ̸= 0, (5)
2
X, for ν1 = µ1 = 0, or ν1 = µ1 = 1.
The second mapping Ψ : X ⇒ ∆2 is specified as follows
Ψ(x) = argmax⟨H(x), ν⟩. (6)
ν∈∆2
where H(x) = (H1 (x), H2 (x)) and
I+ = {i | hi (x) ≥ 0, i = 1, . . . , n}, I− = {i | hi (x) ≤ 0, i = 1, . . . , n}
381
are defined by formulas2
∑
H1 (x) = [ inf hi (x)]+ + xi hih(x)+1
i (x)
, I+ ̸= ∅
i=1,...,m i∈I+
∑
H2 (x) = [ sup hi (x)]− + (1 − xi ) hih(x)−1
i (x)
, I− ̸= ∅.
i=1,...,m i∈I−
If I+ = ∅ or I− = ∅, then by definition H1 (x) = 0 and H2 (x) = 0 respectively. Constructed map is well defined
everywhere except at zero and one for which we postulate
Ψ(0) = (1, 0), Ψ(1) = (0, 1).
Finally, we define the resulting mapping
Υ : X ⇒ X, Υ(x, ν) = Φ(x, ν) × Ψ(x, ν),
which fixed points give us the desired result. The following lemmas describes the important properties of the
mapping Υ(·).
Lemma 1 The mapping Υ : X ⇒ X is a Kakutani map, i.e. it has closed graph and for every κ ∈ X takes
non-empty convex values.
Proof of the lemma is omitted.
Lemma 2 Under the above assumptions, the map φ(·) has nontrivial fixed point in X such that the mass of
the population of each country is nonzero.
The proof of Lemma 2 can be found in [Marakulin, 2016]. Figure 2 illustrates the result of Lemma 2 and following
Theorem 1.
Theorem 1 Let the individual costs be given by (1) and centers be situated on a line parallel to the axis of
abscissa. Then for each positive integer m ∈ N there exists the partition of ABCD into two countries Sleft (x)
and Sright (x), with piecewise linear boundary formed by the points xk , . . . , xl , 1 < k + 1 ≤ l − 1 < m where all
xk+1 , . . . , xl−1 are immigration proof.
Corollary 1 Assume that the costs in formula (1) are calculated relative to the Euclidean distance. Then in the
conditions of Theorem 1 boundary points xk+1 , . . . , xl−1 are suited on classical hyperbola and for a general form
of the metric (e.g. p-norm), these points belong to a generalized hyperbola.
Theorem 2 Let for the rectangle ABCD individual costs be defined by (1) and centers of the country be located
on a line parallel to the axis of abscissa. Then there is immigration proof division into two countries Sleft and
Sright with a continuous border.
It is immaterial fact that the considered area is a rectangular. This result holds for any convex closed bounded
domain. It also follows that the centers can be located on any fixed line (turn the area so that the line is parallel
to the axis of abscissa), and it can be any pair of fixed points.3 Due to the volume of the paper constraints me
omit (more or less standard reasonings via passing to the limits) the proof of this theorem.
1.2 General Partition for a Discontinuous Population Distribution
Following the construction of the first section, consider the rectangular region ABCD, which must be divided
into two countries. Suppose that in this region there is a unit mass of inhabitants whose distribution in contrast
to the previous section is described by the probability Radon measure µ.4 In this case we will not assume the
2 We use standard notations z + = sup{z, 0} and z − = sup{(−z), 0} for any real z.
3 Note that this is one of the fundamental differences between the two-dimensional setting and one-dimensional one.
4 This is an internally regular measure defined on a Borel sigma-algebra. I will also recall that the Borel σ-algebra is specified
by a topology. Set-theoretic operations are performed relative to open subsets, and in the countable number of times, obtaining
different (and only such) elements of algebra. It is important that every continuous function is measurable (integrable) with respect
to the Borel σ-algebra.
382
x1
x2
h(x, y) > 0
x3 = xk
x4
x5
x6 Sright
Sleft centers
x7
h(x, y) < 0 const = −1 < 0
x8 = xl x9 x10
Figure 2: Partition according to (i)–(ii) for const < 0 ⇐⇒ g2 µ(Sleft ) < g1 µ(Sright ).
existence of density for the µ, but we will also postulate that the discrete component has a finite carrier. Indeed,
it is known that for any (nonnegative) Borel measure there is an Lebesgue decomposition in a sum of continuous,
discrete and singular components. According to an interpretation we are interesting in the case µ = ϑ + ν, where
∫
ϑ(B) = h(x)dx, B ∈ B,
B
for some Lebesgue measurable function h : ABCD → R+ and, for an countable family of pairwise distinct
points yk ∈ ABCD, we have
∑
ν({yk }) > 0, k = 1, 2, . . . , ν(B) = ν({yk }), B ∈ B.
k:yk ∈B
Thus, the entire population of the region is divided into two parts—a village one, somehow approximately
uniformly continuously distributed on ABCD and the urban population distributed over not more than a
countable number of points—the population of the k-th city is the value µ({yk }) > 0, k = 1, 2, . . .. Since the
existence of a countable-infinite carrier for ν is untenable from a substantive point of view, we postulate that it
is finite:
card[supp(ν)] < +∞.
In the first section it was proved that for a probabilistic absolutely continuous measure, an immigration-
consistent partition into countries does exist, but what will happen in the general case? In the subsequent
analysis we need some functional spaces and topologies that are relevant in the formulation of the general case.
Indeed, for a compact set K ⊂ R2 , the Radon measures form a simplex in the space ca(K)—it is the space of all
countably additive measures defined on Borel σ-algebra on K. In turn, ca(K) is isomorphic to the space of all
linear continuous functionals over the space of continuous functions C(K), considered with the norm topology
(maximum or uniform convergence). The value of the functional φµ which is associated with the measure µ is
presented by the formula ∫
φµ (f ) = f (x)dµ(x), f ∈ C(K).
Thus we have the right to write [C(K)]′ = ca(K) and consider duality (or pairing) ⟨C(K), ca(K)⟩, in which the
bilinear map ⟨·, ·⟩ (inner product) is given by formula
∫
⟨f, µ⟩ = f (x)dµ(x), f ∈ C(K), µ ∈ ca(K).
For a duality, it is customary to consider and study the various topologies induced by it; the most important
among them are weak topologies. Weak topology can be defined on the source space (here it is C(K)) or conjugate
383
(now it is ca(K)), in the latter case it is usually called weak star topology. By definition, the weak topology is
the weakest locally convex topology for which only functionals from the second duality specify continuous linear
functionals. In our case, σ(C(K), ca(K)) is weak topology on C(K) (by tradition is denoted by σ), is specified
by the space ca(K). A weak star topology σ ∗ (ca(K), C(K)) is defined for ca(K) and is specified via C(K).
Several characterizations can be given for a weak topologies. It is convenient to characterize them in terms of
convergence (a directed family or network). So for σ ∗ (ca([0, 1]), C([0, 1])):
∫ ∫
µξ −→ µ ⇐⇒ ∀g ∈ C(K), g(x)dµξ (x) −→ g(x)dµ(x). (7)
ξ∈Ξ ξ∈Ξ
Weak topologies are often called topologies of pointwise convergence.
The general case of population distribution is analyzed through the passing to the limit for the already proven
case with a continuous density, i. e. I want to find a family fξ ∈ C(K) such that for the measures µ(fξ ) = µξ ,
defined by ∫
µξ (B) = fξ (x)dx, B ∈ B, (8)
B
we would have µξ −→ µ. It is asserted that the space C(K) is dense in ca(K) relative to the weak∗ topology
ξ∈Ξ
σ ∗ (ca(K), C(K)), that is, its closure in σ ∗ gives the whole ca(K) and, therefore, any given measure can be
realized as a limit of measures with continuous densities. Below one can find an (original) short proof, especially
due to we need a little bit different statement: each nonnegative measure to be realized as a limit of measures
with positive continuous densities.
Lemma 3 The set M ⊂ ca(K) of all measures with positive and continuous densities is dense in ca+ (K)5 in
weak∗ topology σ ∗ (ca(K), C(K)).
Proof. Consider the closure of M in σ ∗ (ca(K), C(K)), which we denote by cl∗ (M ). It is clear that
∗
cl (M )⊆ca+ (K) and is convex and weakly∗ closed cone. Suppose the assertion of the lemma is false. Take
ν ∈ ca+ (K) \ cl∗ (M ). We are in the conditions of the classical second separability theorem (strict separability
of closed convex sets, one of which is compact) and we can find a linear functional G, continuous in the weak∗
topology, strictly separating ν from cl∗ (M ), i. e.
∃γ ∈ R : G(cl∗ (M )) ≥ γ > G(ν). (9)
Now for the separating functional the following conclusions can be drawn:
(i) Since the functional is continuous in the weak∗ topology, it must be defined by some continuous function
g ∈ C(K) by formula ∫
G(µ) = g(x)dµ(x), µ ∈ ca(K).
(ii) Since G(cl∗ (M )) ≥ γ, then (by contradiction, M is a cone), we formally conclude
∫
∗
G(cl (M )) ≥ 0 ⇒ g(x)h(x)dx ≥ 0 ∀h ∈ C+ (K) ⇒ g(x) ≥ 0 ∀x ∈ K.
∫
(iii) Due to (ii) and ν ≥ 0 one concludes g(x)dν(x) ≥ 0, that via right hand side of (9) implies γ ≥ 0.
∗
Finally, since 0 ∈ cl (M ), the left∫ inequality in (9) allows us to conclude γ ≤ 0, which together with (iii) gives
γ = 0. However for ν ≥ 0 and γ ≥ g(x)dν(x) ≥ 0 it implies 0 = γ = G(ν) that contradicts to (9).
So it is proven the existence of a family of measures µξ with positive continuous densities fξ , ξ ∈ Ξ which
weakly converges to µ, i. e. (8) and (7) are satisfied. To this family there corresponds a family of curves bξ
defining the inter-country boundary and given by the following equation (with respect to (x, y) ∈ ABCD):
c1 ((x, y), µξ (Sleftξ ), rc (Sleftξ ), zξ ) − c2 (µξ (Sright ξ ), rc (Sright ξ ), zξ ) = 0.
5 It is the cone of all positive elements of the space ca(K)
384
It is assumed that the variable parameters depending on ξ and specifying this equation vary within bounded
limits (in a compact set) and the functions c1 , c2 are continuous. Hence, without loss of generality, we can assume
that
µξ (Sleftξ ) → µlim (Sleft ) = δleft , µξ (Sright ξ ) → µlim (Sright ) = δright ,
(rc (Sleftξ ), zξ ) → (rclim (Sleft ), z lim ).
Further, applying the regularity of the measure µ, one can prove that the family of boundary curves bξ converges
to the limit curve blim , which can be specified by the equation
c1 ((x, y), δleft , rclim (Sleft ), z lim ) − c2 ((x, y), δright , rclim (Sright ), z lim ) = 0.
This is the boundary of the desired inter-country division. Now it is necessary to clarify only one problem.
Everything is obvious in the case when the boundary curve does not pass through any city. However, what if
the city (it is a point of ABCD with nonzero mass) is on the boundary? This city will be divided into two
(unequal) parts, one of which belongs to one country, the other to another one. The proportions of this division
can be found in the following way.
Since the carrier of supp(ν) of the discrete component is finite, there exists ε > 0 such that for each point of
the support a ∈ supp(ν) in the ε-neighborhood Bε (a) there are no other points from the carrier of this point.
Take ε > 0 and consider limξ∈Ξ µξ (Sleftξ \ Bε (m)) = δleft
ε
. Next we find
ε ε a
lim δleft = lim sup δleft = δleft .
ε→+0
This is the mass of the population living in the left country with the exception of “city” a ∈ supp(ν). Thus, the
mass of residents from this city living in the left country can be calculated as δleft − δleft
a a
= νleft . Now in the
right country will live µ(a) − νleft = νright . As a result we have proven the following
a a
Theorem 3 Let the cost functions satisfy the assumptions above and the population distribution is given by
the Radon measure. Then there is an immigration-consistent partition of ABCD into any given number of
countries.
Corollary 2 Under Theorem 2 conditions, the requirement supp(µ) = ABCD can be omitted.
So, we summarize: in the country division of a compact convex area cities (points from a discrete component
of carrier) those are located on the border may have a fractional affiliation (fall under jurisdiction) to the border
countries. Moreover, zero-area countries may emerge. The proper modernization of the concept of immigration-
wealthy division into countries taking into account these circumstances is carried out in a natural way.
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