The Deffuant model on \Z with higher-dimensional opinion spaces

The Deffuant model on with higher-dimensional opinion spaces

Abstract

When it comes to the mathematical modelling of social interaction patterns, a number of different models have emerged and been studied over the last decade, in which individuals randomly interact on the basis of an underlying graph structure and share their opinions. A prominent example of the so-called bounded confidence models is the one introduced by Deffuant et al.: Two neighboring individuals will only interact if their opinions do not differ by more than a given threshold . We consider this model on the line graph and extend the results that have been achieved for the model with real-valued opinions by considering vector-valued opinions and general metrics measuring the distance between two opinion values. As in the univariate case there turns out to exist a critical value for at which a phase transition in the long-term behavior takes place, but depends on the initial distribution in a more intricate way than in the univariate case.

1 Introduction

Consider a simple graph and assume the vertex set to be either finite or countably infinite with bounded maximal degree. The vertices are assumed to represent individuals and each of them is assigned an opinion value. The edges in – being connections between individuals – are understood to embody the possibility of mutual influence. For that reason it is no restriction to focus on connected graphs, as the components could be treated individually otherwise. From different directions including social sciences, physics and mathematics, there has been raised interest in various models for what is called opinion dynamics and deals with the evolution of such a system under a given set of interaction rules. These models are qualitatively different but share similar ideas, see [1] for an extensive survey.
The Deffuant model (introduced by Deffuant et al. [2]) is one of those and features two parameters, the confidence bound and the convergence parameter , shaping the willingness to approach the other individual’s opinion in a compromise. There are two types of randomness in the model: One is the random initial configuration, meaning that at time the vertices are assigned identically distributed opinions, the other are the random encounters thereafter. Serving as a regime for the latter, all the edges in are assigned unit rate Poisson processes, which are independent of one another and the initial configuration. Whenever a Poisson event occurs on an edge, the corresponding adjacent vertices interact in the manner described below. Just like in most of the analyses of this model, we will consider i.i.d. initial opinion values, but comment on how the considerations can be generalized.

By we denote the opinion value at vertex at time . The current value will not change until at some future time a Poisson event occurs at one of the edges incident to , say , which then might cause an update. Let and be the two opinion values of and , just before this happens.

If these opinions lie at a distance less than the confidence bound from one another, they will symmetrically take a step, whose size is scaled by , towards a common compromise, if not they stay unchanged. Although there is a section on vector-valued binary opinions in the original paper by Deffuant et al. [2], using a different model, the Deffuant model with the interaction rule just described was originally only defined for opinions being real-valued and the absolute value as notion of distance. In order to broaden the original scope of this model to vector-valued opinions, the natural replacement for the absolute value is the Euclidean distance

Given this measure of distance, the rule for opinion updates in the Deffuant model reads as follows:

and similarly

Note that choosing gives back the original model.

As the assumptions on the graph force to be countable, there will almost surely be neither two Poisson events occurring simultaneously nor a limit point in time for the Poisson events on edges incident to one fixed vertex. Yet in addition to that there is a more subtle issue in how the simple pairwise interactions shape transitions of the whole system in the infinite setting, putting it into question whether the whole process is well-defined by the update rule (LABEL:dynamics). For infinite graphs with bounded degree, however, this problem is settled by standard techniques in the theory of interacting particle systems, see Thm. 3.9 on p. 27 in [9].

One of the most natural questions in this context – motivated by interpretations coming from social science – seems to be, under what conditions the individual opinions will converge to a common consensus in the long run and under what conditions they are going to split up into groups of individuals holding different opinions instead. In this regard let us define the following types of scenarios for the asymptotic behavior of the Deffuant model on a connected graph as time tends to infinity:

Definition 1
  1. No consensus
    There will be finally blocked edges, i.e. edges s.t.

    for all times large enough. Hence the vertices fall into different opinion groups.

  2. Weak consensus
    Every pair of neighbors will finally concur, i.e.

  3. Strong consensus
    The value at every vertex converges, as , to a common limit , where

    and denotes the distribution of the initial opinion values.

The first analyses of the Deffuant model and similar opinion dynamics were strongly simulation-based and thus confined to a finite number of agents. In [3] for example, Fortunato simulated the long-term behavior of the Deffuant model on four different kinds of finite graphs: Two deterministic examples – the complete graph and the square lattice – as well as two random graphs – those given by the Erdős-Rényi model as well as the Barabási-Albert model. He found strong numerical evidence that, given initial opinions that are independently and uniformly distributed on , a confidence threshold less than leads to a fragmentation of opinions, leads to a consensus – irrespectively of the underlying graph structures that were considered. Later, the simulation studies were extended to the generalization of the Deffuant model to higher-dimensional opinion values, see for instance [10].

There are however crucial differences between the interactions on a finite compared to an infinite graph. In the finite case, statements about consensus or fragmentation tend to be valid not with probability but at best with a probability that is close to : In the standard case of i.i.d.  initial opinions for example, any non-trivial confidence bound, i.e. , can lead to either consensus or fragmentation depending on the initial values and the order of interactions. Furthermore, the fact that the dynamics (LABEL:dynamics) preserves the opinion average of two interacting agents implies that strong consensus follows from weak consensus on a finite graph. This does not have to hold in an infinite setting.

The first major step in terms of a theoretical analysis of the model on an infinite graph was taken by Lanchier [7], who treated the model on the line graph – similarly with an i.i.d.  configuration. His main result implies that there is a phase transition at from a.s. no consensus to a.s. weak consensus. These findings were reproven and slightly sharpened by Häggström [4] to the statement of Theorem 2.1 below, using a non-random pairwise averaging procedure on which he termed Sharing a drink (SAD) to get a workable representation of the opinion values at times .

Using his line of argument, the results were generalized to initial distributions other than by Häggström and Hirscher [5] as well as Shang [12], independently. In [5], the analysis of the Deffuant model was in addition to that extended to other infinite graphs, namely higher-dimensional integer lattices and the infinite cluster of supercritical i.i.d. bond percolation on these lattices.

In this paper we stay on the infinite line graph, that is the integer numbers with consecutive integers forming an edge. The direction in which we want to broaden the analysis is – as already indicated – the generalization of the Deffuant model on to vector-valued opinions. In Section 2, we give a brief summary of the results for real-valued opinions derived in [5], together with the key ideas and tools that were used there.

In Section 3 we establish corresponding results for the case of higher-dimensional opinions sticking, as indicated above, to the Euclidean norm as measure of distance between the opinions of interacting agents. Actually, the main results (Theorem 3.3 and 3.4) in this section match the statement for real-valued opinions (Theorem 2.2) in the sense that the radius of the initial distribution as well as the largest gap in its support – the generalized definitions of which you will find in Definition 2 and 6 – determine the critical value for at which there is a phase transition from a.s. no consensus to a.s. strong consensus. While the concept of a distribution’s radius straightforwardly transfers to higher dimensions, the one of a gap has to be properly redefined and investigated. Doing this, we can in fact characterize the support of the opinion values at times , see Proposition 3.3. Even though we will throughout the paper consider the initial opinions to be i.i.d. it is mentioned in the remark after Theorem 3.4, how the arguments can be extended to particular dependent initial configurations in the way it was done in [5].

Section 4 finally deals with the generalization of the Deffuant model to distance measures other than the Euclidean, in both one and higher dimensions. We pin down properties a general metric (used to determine whether two opinions are close enough to compromise or not) needs to have in order to allow for the results from Section 3 to be preserved (see Theorem 4.5 and 4.6). Examples are given to illustrate the necessity of the requirements imposed on .

At this point it should be mentioned that the vectorial model that was already introduced in the original paper by Deffuant et al. [2] and analyzed quite recently by Lanchier and Scarlatos [8] does not fit the general framework of this paper. Unlike all opinion dynamics considered here, its update rule is different from (LABEL:dynamics) and especially not average preserving, leading to substantial qualitative differences.

2 Background on the univariate case

Theorem 2.1 (Lanchier)

Consider the Deffuant model on the graph , where with i.i.d. unif initial configuration and fixed .

  1. If , the model converges almost surely to strong consensus, i.e. with probability we have: for all .

  2. If however, the integers a.s. split into (infinitely many) finite clusters of neighboring individuals asymptotically agreeing with one another, but no global consensus is approached.

Accordingly, for independent initial opinions that are uniform on , the critical value equals , with subcritical values of leading a.s. to no consensus and supercritical ones a.s. to strong consensus. The case when the confidence bound actually takes on value is still an open problem. The ideas Häggström [4] used to reprove the above result were adapted to accommodate more general univariate initial distributions leading to a similar statement for all such having a first moment , see Thm. 2.2 in [5], which reads as follows:

Theorem 2.2

Consider the Deffuant model on with real-valued i.i.d. initial opinions.

  1. Suppose the initial opinion of all agents follows an arbitrary bounded distribution with expected value and being the smallest closed interval containing its support. If does not lie in the support, let be the maximal, open interval such that lies in and . In this case let denote the length of , otherwise set .

    Then the critical value for , where a phase transition from a.s. no consensus to a.s. strong consensus takes place, becomes . The limit value in the supercritical regime is .

  2. Suppose the initial opinions’ distribution is unbounded but its expected value exists, either in the strong sense, i.e. , or the weak sense, i.e. . Then the Deffuant model with arbitrary fixed parameter will a.s. behave subcritically, meaning that no consensus will be approached in the long run.

The situation at criticality is unsolved with the exception of the case when the gap around the mean is larger than its distance to the extremes of the initial distribution’s support. Given this condition, however, the following proposition (which is Prop. 2.4 in [5]) settles the question about the long-term behavior for critical :

Proposition 2.1

Let the initial opinions be again i.i.d. with being the smallest closed interval containing the support of the marginal distribution, and the latter feature a gap of width around its expected value .

At criticality, that is for , we get the following: If both and are atoms of the distribution , i.e.  and , the system approaches a.s. strong consensus. However, it will a.s. lead to no consensus if either or .

Since the same line of reasoning was used in both [4] and [5] to derive the results we just stated, it is worth taking a closer look on the key concepts involved, especially as they will be the foundation for most of the conclusions drawn in the upcoming sections.

The presumably most central among these is the idea of flat points. If , a vertex is called -flat to the right in the initial configuration if for all :

(2)

It is called -flat to the left if the above condition is met with the sum running from to instead. Finally, is called two-sidedly -flat if for all

(3)

However, in order to understand how vertices being one- or two-sidedly -flat in the initial configuration play an important role in the further evolution of the configuration another concept is indispensable, namely the non-random pairwise averaging procedure Häggström [4] called Sharing a drink (SAD).

Think of glasses being placed at all integers, the one at site being brimful, all others empty. Just as in the Deffuant model, neighbors interact and share, but this time without randomness and confidence bound. In other words, we start with the initial profile , given by and for all , and a finite sequence of edges along which updates of the form (LABEL:dynamics) are performed, i.e. for the profile after step and we get by

(4)

all other values stay unchanged.

Elements of that can be obtained in such a way are called SAD-profiles. The crucial connection to the Deffuant model is that the opinion value at any given time can be written as a weighted average of values at time with weights given by an SAD-profile (see La. 3.1 in [4]). The fact that all SAD-profiles share certain properties (the most important being unimodality) renders it possible to derive characteristics of the future evolution of the Deffuant dynamics given the initial configuration. For instance, the opinion value at a two-sidedly -flat vertex in the initial configuration can never move further than away from the mean (see La. 6.3 in [4]).

These two vital ingredients – flat points and SAD-profiles – of the line of argument in [4] and Sect. 2 in [5] can be adapted in order to analyze the Deffuant model with vector-valued opinions, as we will see in the following section.

3 Deffuant model with multivariate opinions and the Euclidean norm as measure of distance

Having characterized the long-term behavior of the Deffuant dynamics on starting from a general univariate i.i.d. configuration, the next step of generalization with regard to the marginal initial distribution is, as indicated in the introduction, to allow for vectors instead of numbers to represent the opinions. Like in the univariate case, we want the initial opinions to be independent and identically distributed, just now with some common distribution on . This will ensure ergodicity of the setting (with respect to shifts) as before.

In this section we will consider to be equipped with the Borel -algebra generated by the Euclidean norm, denoted by .

Definition 2

If the distribution of has a finite expectation, define its radius by

where denotes the closed Euclidean ball with radius around . Note that the radius of an unbounded distribution is infinite.

The notion of -flatness easily translates to the new setting by just replacing the intervals by balls: If , a vertex is called -flat to the right in the initial configuration if for all :

(5)

similarly for -flatness to the left and two-sided -flatness – compare with (2) and (3).

With these notions in hand we can state and prove a higher-dimensional analogue of Theorem 2.2, valid for initial distributions whose support does not feature a substantial gap around the mean. The proof of this result will be a fairly straightforward adaptation of the methods for the univariate case indicated in Section 2. In contrast, the more general case treated in Theorem 3.4 requires invoking more intricate geometrical considerations.

Theorem 3.3

In the Deffuant model on with the underlying opinion space and an initial opinion distribution we have the following limiting behavior:

  1. If has radius and mass around its mean, i.e.

    (6)

    the critical parameter is , meaning that for we have a.s. no consensus and for a.s. strong consensus.

  2. Let be the random initial opinion vector. If at least one of the coordinates has an unbounded marginal distribution, whose expected value exists (regardless of whether finite, or ), then the limiting behavior will a.s. be no consensus, irrespectively of .

Proof
  1. To show the first part is just like in the univariate case (included in part (a) of Theorem 2.2) little more than following the arguments in the last two sections of [4]: The central arguments go through even for vector-valued opinions as the crucial properties of the absolute value that were used are shared by its replacement in higher dimensions, the Euclidean norm. Because of that, we only sketch the main line of reasoning and refer to Sect. 6 in [4] and Sect. 2 in [5] for a more thorough presentation of the arguments.

    First of all, the (multivariate) Strong Law of Large Numbers – in the following abbreviated by SLLN – tells us that the averages in (5) for large are close to the mean in Euclidean distance. For fixed, choose such that the event

    has positive probability. Using (6) and the fact that the initial opinions are i.i.d., we can locally modify the configuration to conclude that the event has positive probability, implying the -flatness to the right of site – just as it was done in La. 4.2 in [4].

    For , the probability of is non-zero for small enough, hence a vertex can be at distance larger than from initially. Due to the independence of initial opinions, the event that site is -flat to the left, is -flat to the right and has positive probability. Using the SAD representation, it follows – mimicking Prop. 5.1 in [4] – that given such an initial configuration the opinion value at site will be a convex combination of averages in (5) for all times and thus in , due to the convexity of Euclidean balls. The same holds for site and the half-line to the left. Consequently, the edges and will stay blocked for ever. Ergodicity of the initial opinion sequence ensures that with probability (infinitely many) vertices will get isolated that way, which settles the subcritical case.

    In the supercritical regime, i.e. , we focus on two-sidedly -flat vertices: If site is -flat to the left and is -flat to the right, both are two-sidedly -flat – using again the convexity of . By independence this event has positive probability, by ergodicity we will a.s. have (infinitely many) two-sidedly -flat vertices. Mimicking La. 6.3 in [4] literally, we find that vertices which are two-sidedly -flat in the initial configuration will never move further than away from the mean, irrespectively of future interactions. Choosing small, such that  say, will ensure that updates along edges incident to two-sidedly -flat vertices will never be prevented by the distance of opinions exceeding the confidence bound.

    The proof of Prop. 6.1 in [4], which states that neighbors will either finally concur or the edge between them be blocked for large , can be adopted as well: Its central idea – borrowed from physics – that every individual starts with an initial amount of energy that is then partly transferred partly lost in interactions works regardless whether the opinions are shaped by numbers or vectors. Merely in the current setting, the term , that defines the energy at vertex at time , has to be read as a dot product. Again, if the opinions of two neighbors are within the confidence bound but for some fixed , decreases by at least when they compromise. This can not happen infinitely often with positive probability as the expected energy at time is and the expectation of is both non-increasing with and non-negative. For details see Prop. 6.1 and La. 6.2 in [4].

    Following from the considerations above, two-sidedly -flat vertices and their neighbors therefore have to finally concur with probability , forcing the opinion values of the neighbors to eventually lie at a distance strictly less than from the mean as well. By our choice of , this conclusion propagates inductively showing that the limiting behavior will a.s. be strong consensus, if we let tend to .

  2. In order to prove the second claim, we use part (b) of Theorem 2.2, focussing on the th coordinate only. Fix . Since

    a distance of more than in the th coordinate of the opinion vectors for two neighbors implies that the edge between them is blocked. The arguments used for unbounded distributions in Theorem 2.2 (see Thm. 2.2 in [5]) show that under the given conditions, there are a.s. vertices that differ more than from both their neighbors in the th coordinate (with respect to the absolut value) in the initial configuration and this will not change no matter whom their neighbors will compromise with. Consequently, the corresponding opinion vectors will always be at Euclidean distance more than .

Remark

Pretty much as in the univariate setting, the case where all unbounded coordinates of do not have an expected value (neither finite nor nor ) remains unsolved by Theorem 3.3.

When it comes to bounded initial distributions which do have a large gap around the mean, the picture in higher dimensions drastically changes – something that

will require several preliminary results before we are ready to state and prove this section’s main result, Theorem 3.4. The major difference to the univariate case is that with higher-dimensional opinions the update along some edge can actually lead to a situation, where both and come closer to the opinion corresponding to a third vertex , which lies within the confidence bound of neither nor , see the picture on the right.

In the case of real-valued opinions this is impossible, because in that setting an update along always increases , if does not lie in between and .

To illustrate how this changes the conditions, let us consider the initial distributions , where denotes the Euclidean unit sphere in . For this is just , which by Theorem 2.2 has the trivial critical value . For however, the fact that opinions close to each other can compromise in order to form a central opinion will bring down to the radius 1 of the distribution as we will see in the sequel.

The statement of the main result in this section, Theorem 3.4, resembles very much the one of Theorem 2.2 (a), only the notion of a gap in the initial distribution has to be reinterpreted in the higher-dimensional setting, making the proof of this generalized result rather technical. However, while establishing auxiliary results, we will gain additional information about the set of opinion values that can occur in the Deffuant model at times depending on the initial distribution and the confidence bound. When it comes to the initial distribution , the most important features besides its expected value are its support and the corresponding radius.

Definition 3

Consider an -valued random variable . Its support is the following subset of , which is closed with respect to the Euclidean metric:

Observe that this definition corresponds to the standard notion of spectrum of a measure (see for example Thm. 2.1 and Def. 2.1 in [11]) – applied to the distribution of a random variable.

If the initial distribution has a finite expectation, the radius can also be written as

as the following proposition shows.

Proposition 3.2

If , we have

(7)

Proof

First, consider a set which is compact in and a subset of the complement of . The claim is that these properties of imply . Indeed, for every there exists s.t. . Let denote the open Euclidean ball with radius around , then is an open cover of , which by compactness has a finite subcover . Consequently

If is greater than the supremum in (7) it follows that . Since

and the right-hand side is a countable union of nullsets with respect to , we get , which means that is greater or equal to the infimum in (7).

On the other hand, if is less than the supremum, there exists a point , which consequently has a positive distance to the closed ball . This gives

In other words, does not appear in the set the infimum is taken over. Putting both arguments together proves (7).

Definition 4
  1. For a finite graph and an edge let the update described in (LABEL:dynamics), considered as a deterministic map on the set of -valued profiles, be denoted by . So if is applied to it just means that all values stay unchanged with the only exception of

    (8)
  2. Consider a finite section of the line graph, a finite sequence of edges and some values in . Such a triple will from now on be called a finite configuration.
    To update the configuration (with respect to ) will mean that we take as initial opinions, i.e. we set for all , and then apply to .

    Slightly abusing the notation, let the outcome, i.e. the final opinion values , be denoted by .

  3. Let denote the initial distribution . For , let denote the set of vectors in which the opinion values of finite configurations can collectively approach, if updated according to confidence bound . More precisely, if and only if for all , there exist some , and as above, such that updating the configuration with respect to yields for all .

It is worth emphasizing that finite configurations are supposed to mimick the dynamics of the Deffuant model, interpreting as the locations of the first Poisson events on the edges in (strict) chronological order. In this respect, considering , we can choose the sequence such that only Poisson events causing an actual update are considered by simply eliminating all events on edges where the opinions of the two vertices are more than apart.

Note that according to the definition, depends on and , as well as , the latter being less obvious. See Example 3.4 below for an instance where actually makes a difference. Let us now turn to various properties of the set .

Lemma 3.1

Fix the distribution of and let and be defined as above.

  1. is closed and increases with .

  2. for all , where denotes the convex hull, the closure of a set .

Proof
  1. The first claim follows directly from the definition: For a sequence in such that and every , there exists some . Due to , there exists a finite configuration with all final opinion values in . But since , this implies .

    As for the second claim, since we are free to choose the edge sequence in finite configurations, it is obvious that making larger only allows for more options when we are to come up with a setting that brings the opinion values collectively inside for some given and .

  2. The first inclusion is trivial, as for the finite configuration with will do. The second inclusion is due to the fact that every update of opinions is a convex combination, see (8). Consequently, all final opinion values of finite configurations lie within . The last inclusion, which is meaningful only for , follows from Proposition 3.2 and the fact that is both convex and closed.

It should be mentioned that an easy corollary to Carathéodory’s Theorem on the convex hull states that the convex hull of a compact set in is compact as well. If has a bounded support, this implies that the convex hull of is actually closed, i.e. .

Example 3.1

To get familiar with the idea behind , let us consider the discrete real-valued initial distribution given by . It is not hard to see that this implies . Having the Taylor expansion of the logarithm in mind we find

By Theorem 2.2 we get , since and the largest gap in between the point masses is .

For two point masses situated at and at distance , all convex combinations of are in : For and , take s.t.

Let us set up a finite configuration with vertices, and as well as enough Poisson events on every edge (in an appropriate order) such that – having updated the configuration according to the edge sequence – the outcome will be at distance less than from the average for all . Since all the opinion values lie in an interval of length at most in the beginning and hence always will, we could choose the edge sequence by always taking the edge with largest current discrepancy next, to see that a finite sequence with the claimed property exists. This will ensure

hence . This observation together with the fact that gaps of width larger than can not be bridged leads to

where .

Lemma 3.2
  1. For all and , the set is convex.

  2. If , then .

  3. The connected components of are convex and at distance at least from one another. If is connected, then .

  4. If and has mass around its mean, i.e. condition (6) holds, then already for .

  5. For , the set-valued mapping

    is piecewise constant with only finitely many jumps on for all .

  6. If is connected and finite, then

Proof
  1. The proof of the first part of this lemma follows the idea of the above example. Let and their distance be . Let . For any , there exist finite configurations and with final values in and respectively. For choose again s.t.

    We define a new finite configuration by putting copies of and copies of next to each other: Their finite sections of the line graph (together with the assigned initial values) will be concatenated blockwise – the order among the blocks being irrelevant – by adding an edge between two consecutive blocks in order to form the underlying line graph of a larger finite configuration. To get an edge sequence for the whole configuration we will simply string together the edge sequences of the individual copies, again in a blockwise manner and arbitrary order.

    Updating according to the edge sequence will then bring all the opinion values within distance of one another. Therefore, we can bring the final outcomes arbitrarily close, say at distance at most , to the average of the initial values, let’s denote it by , by just adding a large enough (but finite) number of Poisson events on each edge (appropriately ordered as before). From the properties of the chosen building blocks, and , it readily follows that the initial average is at distance at most from . This entails for every vertex of the finite configuration

    which shows .

  2. By Lemma 3.1 it is enough to show . Thus, letting , we have to show that . But since can be at most , this is done as described in Example 3.1, just the line segment plays now the role of the interval considered there.

  3. First of all, the connected components of are actually path-connected and moreover the pathes can be chosen to be polygonal chains: Assume that a connected component contains more than one path-connected component. Fix one such, say . Due to connectedness of , a second one must exist s.t. the Euclidean distance between and is . But part (a) then implies that also is path-connected, a contradiction. Moreover, using the statement of part (a) we can transform any curve in to a polygonal chain which completely lies in .

    Let us turn to the convexity of connected components. Fix a component of and , s.t. , since otherwise (a) guarantees

    By the above, there exists a polygonal chain in , say

    such that and is continuous and piecewise linear. Let us define where , if and otherwise. Using (a) and these intermediate points shows that we can assume without loss of generality a certain sparseness of the chain, namely that its intermediate points are s.t. pairwise distances in are at least and hence , where denotes the length of the original chain. Note that the modification of the polygonal chain as just described will only decrease its length.

    Given a polygonal chain in connecting and , let us assume that the minimal angle at an intermediate point is at . Considering and using (a) once more, we can replace by the two intersection points of the ball’s boundary and the chain and conclude that the polygonal chain through the points still lies in and is at least by shorter.

    We can then sparsify the updated chain as described above and denote the result by . Iterating the whole procedure gives a sequence of shorter and shorter polygonal chains in connecting and . Since the length is bounded below by , the internal angels must approach uniformly. Let be the angles at . An easy geometric argument yields that all points on the chain are at distance at most

    from the line through and , if , as for all . This also holds for the endpoint , which is why the maximal distance of a point on the chain to the line segment between and is bounded by . Let and correspond to . Then

    implies that the sequence must approach the line segment between and , i.e. , uniformly – in the sense that

    Since being a component of is closed, we find which proves the convexity of C.

    Assuming that there are two points in different connected components, say s.t. , already implies (by part (a)) that is connected, as before. Finally, if is connected, what we just proved induces that it is convex. Being a closed superset of , this implies

    which by Lemma 3.1 is all that needed to be shown.

  4. Let us now assume that has not only a finite radius but also mass around its mean, that is . For , is then connected, which by part (c) implies the claim. Indeed, let and choose a point in . By the choice of , all points in are at distance less than from , which by the reasoning in part (a) and (see Lemma 3.1) implies for all , hence the connectedness of .

  5. The first thing to notice is that, given , for all the set has finitely many connected components. Indeed, choose a point in each, then the open balls must be disjoint by (c) and lie within . Consequently, there can’t be more than of them.

    Let be the connected components of , for some , and the minimal distance between them. When is made larger than , at least two of the components merge. Hence there can be only further jumps. For we have