On the singularities of distance functions in Hilbert spaces

Abstract

For a given closed nonempty subset E of a Hilbert space H, the singular set $\Sigma _E$ consists of the points in $H\setminus E$ where the distance function $d_E$ is not Fréchet differentiable. It is known that $\Sigma _E$ is a weak deformation retract of the open set $\mathcal {G}_E=\{x\in H: d_{\overline{{\text {co}}}\,E}(x)< d_E(x)\}$. This short paper sheds light on the relationship between the connected components of the three sets $\Sigma _E\subset \mathcal {G}_E\subseteq H{\setminus } E$.

1 Introduction

This note is devoted to the singularities of the distance function of an arbitrary closed nonempty subset E of a real Hilbert space H. The singular set $\Sigma _E$ consists of all points in $\complement E= H\setminus E$ where the distance function

$$\begin{aligned} d_E(x)=\inf _{y\in E}\Vert x-y\Vert \end{aligned}$$

fails to be Fréchet differentiable. Along with the distance function, the metric projection onto E carries each $x\in H$ to

$$\begin{aligned} P_E(x)= \{y\in E:\Vert x-y\Vert =d_E(x)\}. \end{aligned}$$

A point $x\in H$ is nonsingular if and only if the multivalued mapping $P_E: H\rightrightarrows H$ is single-valued and continuous at x [10]. The set of nonsingular points $\complement \Sigma _E$ is a dense $G_\delta$ set for $d^2_E-\Vert \cdot \Vert ^2$ is a concave function [2, 3]. The singular set is an intriguing object, in general; e.g., $\Sigma _E$ is dense in $\complement E$ for most compact sets $E\subset H$ when H is a separable Hilbert space, in particular, when $\dim H<\infty$ [9, 14, 18]. It is noteworthy that this topic is linked to partial differential equations set in Hilbert spaces; namely, $d_E(x)$ is a viscosity solution of the eikonal equation $\Vert dw(x)\Vert =1$ in $\complement E$. Furthermore, when $\dim H<\infty$, the singular set coincides with the medial axis or ambiguous locus of E which is the set of all points having more than one nearest point in E. In ${\mathbb {R}}^2$ or ${\mathbb {R}}^3$, the medial axis can be used as a device for shape recognition [4, 12].

A source of inspiration for this short contribution has been the paper [8] which, among other things, characterizes the unbounded connected components of $\Sigma _E$ in a finite-dimensional framework drawing on the one-to-one correspondence discovered in [17] between the components of $\Sigma _E$ and those of $\overline{{\text {co}}}\,E{\setminus } E$. Here, $\overline{{\text {co}}}\,E$ signifies the closed convex hull of E, i.e., the intersection of all closed convex supersets of E. At the same time, in a general Hilbert space, the research presented in [16] has revealed that, in a sense, the singularities are generated in the open set

$$\begin{aligned} \mathcal {G}_E=\{x\in H: d_{\overline{{\text {co}}}\,E}(x)<d_E(x)\} \end{aligned}$$

and that $\mathcal {G}_E$ can be continuously shrunk onto $\Sigma _E$. It is elementary to show that

$$\begin{aligned} \overline{{\text {co}}}\,E\setminus E\subseteq \mathcal {G}_E\subseteq \complement E. \end{aligned}$$

In fact, $\overline{{\text {co}}}\,E\setminus E$ is a strong deformation retract of $\mathcal {G}_E$ where the deformation retraction is given by

$$\begin{aligned} \Phi (\lambda ,x)=(1-\lambda )x+\lambda P_{\overline{{\text {co}}}\,E}(x) \end{aligned}$$

for all $(\lambda ,x)\in [0,1]\times \mathcal {G}_E$. Here, as we recall, the metric projection onto $\overline{{\text {co}}}\,E$ is a nonexpansive mapping. We shall examine the inclusions $\Sigma _E\subset \mathcal {G}_E\subseteq \complement E$ with the objective of finding the relationship between the connected components of these sets in a general Hilbert space setting. The principal result of [16] is the following highly relevant statement.

Theorem 1

Let E be a closed nonempty subset of a real Hilbert space H. Then, $\Sigma _E\subset \mathcal {G}_E$ and $\Sigma _E$ is a weak deformation retract of $\mathcal {G}_E.$ In particular, the inclusion $\Sigma _E\subset \mathcal {G}_E$ is a homotopy equivalence.

Similar results on the homotopy type of the singular set in the setting of complete Riemannian manifolds appear in [1, 7].

Corollary 1

There exists a bijection between the family of connected components of $\mathcal {G}_E$ and the family of connected components of $\Sigma _E.$

• (i) Each connected component G of $\mathcal {G}_E$ contains a unique connected component $S_G$ of $\Sigma _E$ while each connected component S of $\Sigma _E$ is contained in a unique connected component $G_S$ of $\mathcal {G}_E.$ Moreover, $S_G=G\cap \Sigma _E.$

• (ii) Each connected component G of $\mathcal {G}_E$ is open in H and path connected while each connected component S of $\Sigma _E$ is relatively open and path connected.

Remark 1

By part (ii) of this corollary, in either space, the connected components coincide with the path components. Therefore, we shall simply write “components” in the sequel. This remark applies to the components of $\complement E$ as well.

From algebraic topology, we know that the homotopy groups $\pi _n(X,x_0)$ are prominent examples of homotopy invariants [13]. The fundamental group $\pi _1(X,x_0)$ paints an algebraic image of a topological space X from the loops in the space, the loops being paths whose initial and terminal points are the same, namely the base point $x_0\in X$. The group operation of $\pi _1(X,x_0)$ is given by concatenation of loops. In general, the elements of $\pi _n(X,x_0)$ are homotopy classes of base point preserving continuous mappings from the n-sphere to X. When X is path connected, the choice of base point is immaterial and it is customary to abbreviate the notation of the homotopy group to $\pi _n(X)$. If two spaces X and Y are homotopy equivalent, and one is path connected, then both spaces are actually path connected and their homotopy groups $\pi _n(X)$ and $\pi _n(Y)$ are isomorphic [13, Cor. 11.27] for every $n\in {\mathbb {Z}}_+$.

Corollary 2

Let G be a component of $\mathcal {G}_E.$ Then the component $S_G=G\cap \Sigma _E$ of $\Sigma _E$ is a weak deformation retract of G. In particular, the n-th homotopy groups $\pi _n(S_G)$ and $\pi _n(G)$ are isomorphic for every $n\in {\mathbb {Z}}_+.$

As a special case, we note that $S_G$ is simply connected if and only if G is simply connected. Indeed, a path connected space X is simply connected exactly when its fundamental group $\pi _1(X)$ is trivial.

The organization of the rest of the paper is as follows. In the next section, the main result of this note is stated, in Theorem 2, and illustrated. The third and last section furnishes a sequence of statements about the relationships between the components of the sets of the inclusions $\Sigma _E\subset \mathcal {G}_E\subseteq \complement E$, which lead to a proof of Theorem 2.

2 Statement of the principal results

The ray that emanates from $z_0$ and passes through another point $x_0$ is given by

$$\begin{aligned} \overrightarrow{z_0x_0}=\{(1-\lambda )z_0+\lambda x_0: \lambda \in [0,\infty )\}. \end{aligned}$$

For any initial point $(t_0,x_0)\in (0,\infty )\times H$, the intrinsic characteristic $\varvec{x}(t)$ is defined for $t\in [t_0,\infty )$ by $\varvec{x}(t_0)=x_0$ and

$$\begin{aligned} \{\varvec{x}(t)\}=\mathop {\textrm{arg}\,\textrm{max}}\limits _{x\in H} \left( \frac{1}{2t}d_E^2(x)-\frac{1}{2(t-t_0)}\Vert x_0-x\Vert ^2 \right) \quad \text {if}\quad t\in (t_0,\infty ). \end{aligned}$$

(1)

The element $\varvec{x}(t)\in H$ is well-defined by virtue of the fact that the right-hand side of (1) is a uniformly concave function of $x\in H$ due to the concavity of $x\mapsto d^2_E(x)-\Vert x\Vert ^2$. This fact ensures that there exists a unique maximizer in (1). Indeed, owing to the Hilbert space setting, expanding $\Vert x-y\Vert ^2$, we find that

$$\begin{aligned} d^2_E(x)=\Vert x\Vert ^2-2A(x)\quad \text {for all}\quad x\in H, \end{aligned}$$

(2)

where Asplund’s function [3] is the continuous convex function $A: H\rightarrow {\mathbb {R}}$ that sends each $x\in H$ to

$$\begin{aligned} A(x)=\sup _{y\in E}\left( \langle x,y\rangle -\Vert y\Vert ^2/2\right) . \end{aligned}$$

One notices a relationship between (1) and the Lasry–Lions regularization technique [11].

Corollary 1 asserts that the components of the singular set are given by $S_G=G\cap \Sigma _E$ where G runs through all components of $\mathcal {G}_E$. We proceed to the statement of our main result which further clarifies the relationship between the components of the three spaces $\Sigma _E\subset \mathcal {G}_E\subseteq \complement E$. The following three cases, (i), (ii), and (iii), are clearly mutually exclusive and exhaustive.

Theorem 2

Let E be a closed nonempty proper subset of H. Assume that C is a component of $\complement E.$

• (i) Suppose that C is bounded. Then C is a component of $\mathcal {G}_E$ as well and $C\subseteq \overline{{\text {co}}}\,E{\setminus } E.$ Moreover, $C\cap \Sigma _E\ne \emptyset$ is connected too, i.e., it is the unique component of $\Sigma _E$ that intersects C.

• (ii) If C is unbounded and disjoint from $\overline{{\text {co}}}\,E,$ then $C\cap \mathcal {G}_E=\emptyset$ and C is absent of singularities.

• (iii) Assume that C is unbounded and $C\cap \overline{{\text {co}}}\,E\ne \emptyset .$ Then C includes at least one component G of $\mathcal {G}_E.$ Every such component $G\subseteq C$ is unbounded and $S_G=G\cap \Sigma _E$ is unbounded as well. We may in fact distinguish between two subcases:

  • (a) If $C\setminus \overline{{\text {co}}}\,E\ne \emptyset ,$ then each component $G\subseteq C$ of $\mathcal {G}_E$ satisfies $G\setminus \overline{{\text {co}}}\,E\ne \emptyset$ and G includes rays. In fact, if $x_0\in G\setminus \overline{{\text {co}}}\,E$ and $P_{\overline{{\text {co}}}\,E}(x_0)=\{z_0\},$ then $z_0\ne x_0$ and the ray $\overrightarrow{z_0x_0}$ from $z_0$ through $x_0$ is included in G. Moreover, the intrinsic characteristic $\varvec{x}(t)$ emanating from $x_0$ is unbounded, $\varvec{x}(t)\in G$ for all t, and $\varvec{x}(t)\in S_G=G\cap \Sigma _E$ for all sufficiently large values of t.

  • (b) If $C\subseteq \overline{{\text {co}}}\,E,$ then C is itself a component of $\mathcal {G}_E$ and $C\cap \Sigma _E$ is unbounded.

This theorem may be viewed as a Hilbert space version of [8, Thm. 3.1]. The analysis conducted in [8] lies in the framework of closed sets $E\subset {\mathbb {R}}^n$ and, instead of $\mathcal {G}_E$, $\overline{{\text {co}}}\,E\setminus E$ plays a main role. We next illustrate Theorem 2 recalling that, when $\dim H<\infty$, a point x is singular if and only if it has more than one nearest point in E.

Example 1

Let E be the closed set in $H={\mathbb {R}}^2$ depicted as

• (a) E along with the components, B and C, of $\complement E$ where B is the bounded one inside the semicircle.

• (b) The singular set $\Sigma _E$ is composed of straight line segments and pieces of parabolas. The sets $\mathcal {G}_E$ and $\Sigma _E$ have three components each, namely $G_k$ and $S_{G_k}=G_k\cap \Sigma _E$ for $k\in \{1,2,3\}$, respectively, where $G_1=B$ while $G_2$ and $G_3$ are proper open unbounded subsets of C containing rays. Unlike $S_{G_1}$, the components $S_{G_2}$ and $S_{G_3}$ of the singular set are unbounded.

In subcase (b) of alternative (iii) of Theorem 2, i.e., when $C\subseteq \overline{{\text {co}}}\,E$, $G=C$ need not include a ray.

Example 2

In $H={\mathbb {R}}^2$, let $\mathcal {S}$ denote the Archimedean spiral defined in polar coordinates by $r=\theta$, $\theta \in [0,\infty )$. The curvature $\kappa (\theta )=(\theta ^2+2)/(\theta ^2+1)^{3/2}$ of $\mathcal {S}$ is a decreasing function of $\theta \in (0,\infty )$ and $\kappa (0+)=2$. Since the curvature is bounded, for a sufficiently small $\delta >0$, $\mathcal {S}_\delta :=\mathcal {S}+D(0,\delta )$ is an open tubular neighborhood of $\mathcal {S}$. Here, $D(0,\delta )$ stands for the open disc of radius $\delta$ centered at the origin. Let $E=\complement \mathcal {S}_\delta$. In this case, ${\text {co}}E=H$, $G=C=\complement E$ while $\Sigma _E=\mathcal {S}$. Indeed, $P_E(0)$ is a semicircle of radius $\delta$ and $P_E(x)$ has two elements for every other $x\in \mathcal {S}$. Thus $\mathcal {S}\subseteq \Sigma _E$. Suppose that some $x\in \complement E{\setminus }\mathcal {S}=\mathcal {S}_\delta {\setminus }\mathcal {S}$ has two nearest points $y_1$ and $y_2$ in E. Then $x-y_1$ and $x-y_2$ are nonparallel normal vectors to the boundary of E such that the line segment $[y_k,x]$ and $\mathcal {S}$ are disjoint for $k=1,2$ (indeed, $[y_k,x)\cap \Sigma _E=\emptyset$ as proved in, e.g., [15, Prop. 5]). However, this is ruled out by our choice of $\mathcal {S}_\delta$ showing that $\Sigma _E=\mathcal {S}$.

3 The components of $\complement E$, $\mathcal {G}_E$, and $\Sigma _E$ and the proof of Theorem 2

The intrinsic characteristics defined by (1) play a pivotal role. The virtue of these arcs is that they propagate singularities, which is manifested in the following dichotomy [15, Thm. 7]: either

• (i) $\varvec{x}(\cdot )$ is nonsingular, i.e., $\varvec{x}(t)\notin \Sigma _E$ for all $t\in [t_0,\infty )$; or

• (ii) $\varvec{x}(\cdot )$ is eventually singular in the sense that there exists a cut time $t^*\in [t_0,\infty )$ such that $\varvec{x}(t)\notin \Sigma _E$ for all $t\in [t_0,t^*)$ but $\varvec{x}(t)\in \Sigma _E$ for all $t\in (t^*,\infty )$.

The cut point $\varvec{x}(t^*)$ belongs to the closed hull $\overline{\Sigma }_E$ and may or may not belong to $\Sigma _E$ itself. Another important property is the Lipschitz continuity of $(t,x_0)\mapsto \varvec{x}(t,x_0)$ on $[t_0,\infty )\times H$ [15, Prop. 3]. Assuming $x_0\notin \Sigma _E$, $P_E(x_0)$ is a singleton $\{z_0\}$ and the motion is rectilinear with velocity $u_0=(x_0-z_0)/t_0$, i.e.,

$$\begin{aligned} \varvec{x}(t)=z_0+tu_0=x_0+(t-t_0)u_0 \end{aligned}$$

as long as $\varvec{x}(t)$ remains nonsingular (consult Proposition 8 and Theorem 12 in [15]). Clearly, $u_0$ is a generalized normal vector at the boundary point $z_0$. The notion of intrinsic characteristics was introduced in [5] and has also been utilized in [6, 7, 15] in order to study singular propagation for Hamilton–Jacobi equations.

As regards the basic role of $\mathcal {G}_E$ in the generation of singularities, the assertions of the following two propositions were demonstrated in [16].

Proposition 1

The following conditions are equivalent for any point $x_0\in H:$

• (i) The intrinsic characteristic issuing from $x_0$ is eventually singular;

• (ii) $x_0\in \mathcal {G}_E;$

• (iii) $P_{\overline{{\text {co}}}\,E}(x_0)\cap E=\emptyset .$

Proposition 2

Suppose that $x_0\in \mathcal {G}_E$ and let G be the component of $\mathcal {G}_E$ that $x_0$ lies in. Then the following assertions are true for the intrinsic characteristic emanating from $x_0.$

• (i) $\varvec{x}(t)\in G$ for all $t\in [t_0,\infty )$ and $\varvec{x}(t)\in S_G=G\cap \Sigma _E$ for all sufficiently large t.

• (ii) Assume in addition that $x_0\notin \overline{{\text {co}}}\,E.$ Then $d_E(\varvec{x}(t))$ and $\Vert x_0-\varvec{x}(t)\Vert$ are unbounded nondecreasing functions of $t\in [t_0,\infty );$ in particular, G and $S_G$ are unbounded sets. Furthermore, writing $P_{\overline{{\text {co}}}\,E}(x_0)=\{z_0\},$ $z_0\ne x_0$, and the ray $\overrightarrow{z_0x_0}$ from $z_0$ through $x_0$ is included in G.

We have already stated in Theorem 1 that $\Sigma _E$ is a weak deformation retract of $\mathcal {G}_E$. The intrinsic characteristics constitute a prime tool in the proof of Theorem 1 obtained in [16]. We turn our attention to the demonstration of Theorem 2. The singular set is empty if and only if E is a convex set. As regards the presence or absence of singularities in a given connected component of $\complement E$ the following can be said.

Lemma 1

The following conditions are equivalent for any component C of $\complement E:$

• (i) $C\cap \overline{{\text {co}}}\,E\ne \emptyset ;$

• (ii) $C\cap \mathcal {G}_E\ne \emptyset ;$

• (iii) $C\cap \Sigma _E\ne \emptyset .$

Proof

The implication (i) $\Rightarrow$ (ii) is correct because

$$\begin{aligned} C\cap \overline{{\text {co}}}\,E=C\cap ( \overline{{\text {co}}}\,E\setminus E) \subseteq C\cap \mathcal {G}_E. \end{aligned}$$

“(ii) $\Rightarrow$ (iii)”: Assume (ii). Then there exists at least one component G of $\mathcal {G}_E$ such that $G\subseteq C$. On account of Corollary 1, $S_G=G\cap \Sigma _E$ is a component of $\Sigma _E$. Thus $S_G\subseteq C\cap \Sigma _E$ and (iii) is fulfilled.

“(iii) $\Rightarrow$ (i)”: Assuming $x_0\in C\cap \Sigma _E$, let G be the component of $\mathcal {G}_E$ that contains $x_0$. The point $z_0=P_{\overline{{\text {co}}}\,E}(x_0)$ belongs to the same component G by part (ii) of Proposition 2 and clearly $G\subseteq C$. Therefore, $z_0\in C\cap \overline{{\text {co}}}\,E$ and (i) is satisfied. $\square$

Lemma 2

Let C be a component of $\complement E.$ Then the implications (i) $\Rightarrow$ (ii) $\Rightarrow$ (iii) hold between the following conditions:

• (i) C is bounded;

• (ii) $C\subseteq \overline{{\text {co}}}\,E\setminus E;$

• (iii) C is a component of $\mathcal {G}_E$ as well.

Proof

“(i) $\Rightarrow$ (ii)”: By the Hahn–Banach separation theorem, each $x\notin \overline{{\text {co}}}\,E$ lies in a half-space that does not meet $\overline{{\text {co}}}\,E$ hence in an unbounded component of $\complement E$. We conclude by contraposition.

“(ii) $\Rightarrow$ (iii)”: Let (ii) be fulfilled. Since $\overline{{\text {co}}}\,E{\setminus } E\subseteq \mathcal {G}_E$, C is also a component of $\mathcal {G}_E$. $\square$

Lemma 3

Suppose that G is a component of $\mathcal {G}_E$ and let C be the component of $\complement E$ that includes G. The following conditions are then equivalent:

• (i) $S_G=G\cap \Sigma _E$ is unbounded;

• (ii) G is unbounded;

• (iii) C is unbounded and intersects $\overline{{\text {co}}}\,E$.

Proof

The implication (i) $\Rightarrow$ (ii) is trivial.

“(ii) $\Rightarrow$ (i)”: Arguing by contradiction, suppose that G is unbounded and that $S_G=G\cap \Sigma _E$ is a subset of the open ball B(0, R) for some radius $R>0$. Fix a point $y_0\in E$ and select next a point $x_0\in G$ such that

$$\begin{aligned} \Vert x_0\Vert >\Vert y_0\Vert +2R. \end{aligned}$$

(3)

The intrinsic characteristic $\varvec{x}(\cdot )$ starting at time $t_0>0$ from the point $x_0$ is eventually singular and never leaves G by Proposition 2. The cut point $\varvec{x}(t^*)$ is an element of $\overline{\Sigma }_E$ and therefore $t^*>t_0$; otherwise, $x_0=\varvec{x}(t_0)=\varvec{x}(t^*)\in \overline{\Sigma }_E$ contradicting that a neighborhood of $x_0$ is absent of singularities. Since $x_0$ is nonsingular, $P_E(x_0)$ is a singleton $\{z_0\}$ and in setting $u_0=(x_0-z_0)/t_0$, we have $\varvec{x}(t)=z_0+tu_0$, $\Vert u_0\Vert =d_E(x_0)/t_0$, and $P_E(\varvec{x}(t))=\{z_0\}$ for all $t\in [t_0,t^*)$ on account of Proposition 8 and Theorem 12 in [15]. We deduce that

$$\begin{aligned} \Vert \varvec{x}(t)-x_0\Vert =(t-t_0)\Vert u_0\Vert <t\Vert u_0\Vert =\Vert \varvec{x}(t)-z_0\Vert =d_E(\varvec{x}(t))\le \Vert \varvec{x}(t)-y_0\Vert \end{aligned}$$

(4)

for all $t\in [t_0,t^*)$. Similarly as in the proof of [8, Thm. 3.1], we infer from (4) that

$$\begin{aligned} \Vert \varvec{x}(t)\Vert&\ge \Vert x_0\Vert -\Vert \varvec{x}(t)-x_0\Vert \ge \Vert x_0\Vert -\Vert \varvec{x}(t)-y_0\Vert \\&\ge \Vert x_0\Vert -\Vert \varvec{x}(t)\Vert -\Vert y_0\Vert \end{aligned}$$

for all $t\in [t_0,t^*)$ and so

$$\begin{aligned} 2\Vert \varvec{x}(t^*)\Vert \ge \Vert x_0\Vert -\Vert y_0\Vert . \end{aligned}$$

(5)

Thus $\Vert \varvec{x}(t^*)\Vert >R$ owing to (5) and (3) contradicting the assumption that $G\cap \Sigma _E\subset B(0,R)$ in view of $\varvec{x}(t^*)\in G\cap \overline{\Sigma }_E$.

“(ii) $\Rightarrow$ (iii)”: Assume (ii). Then C is clearly unbounded and $z_0=P_{\overline{{\text {co}}}\,E}(x_0)$ belongs to G for any $x_0\in G$ by virtue of Proposition 2. In particular, $z_0\in C\cap \overline{{\text {co}}}\,E$ and so (iii) is met.

“(iii) $\Rightarrow$ (ii)”: Assume (iii). If $C\subseteq \overline{{\text {co}}}\,E,$ then $G=C$ by Lemma 2 and so (ii) is satisfied. We next consider the case where $C\setminus \overline{{\text {co}}}\,E\ne \emptyset$. It suffices to establish that $G\setminus \overline{{\text {co}}}\,E\ne \emptyset$ for then G is unbounded by Proposition 2. Assume on the contrary that $G\subseteq \overline{{\text {co}}}\,E$ and pick a pair of points $x_0\in G\subseteq \overline{{\text {co}}}\,E$ and $x_1\in C\setminus \overline{{\text {co}}}\,E$ together with a path $\xi$ in C from $x_0$ to $x_1$, i.e., a continuous $\xi :[0,1]\rightarrow C$ such that $\xi (0)=x_0$ and $\xi (1)=x_1$. Along this path, there is a boundary point of $\overline{{\text {co}}}\,E$; indeed, if

$$\begin{aligned} c=\inf \{t\in [0,1]: \xi (t)\notin \overline{{\text {co}}}\,E \}, \end{aligned}$$

then $c\in (0,1)$ and $\xi (c)$ lies on the boundary of the closed set $\overline{{\text {co}}}\,E$. We note that $\xi (c)\in G$ because $\xi (t)\in \overline{{\text {co}}}\,E{\setminus } E\subseteq \mathcal {G}_E$ for all $t\in [0,c]$ and $\xi (0)\in G$. It ensues that $\xi (t)\in G\setminus \overline{{\text {co}}}\,E$ for some $t>c$ since G is open, which violates the assumption that $G\subseteq \overline{{\text {co}}}\,E$. $\square$

We are now in the position to conclude the proof of our main result.

Proof of Theorem 2

Assertion (i) is covered by Corollary 1 and Lemma 2 while (ii) follows from Lemma 1.

(iii) The hypothesis that C and $\overline{{\text {co}}}\,E$ intersect ensures that C includes a component G of $\mathcal {G}_E$ on account of Lemma 1. If actually $C\subseteq \overline{{\text {co}}}\,E,$ then $G=C$ and $S_G=C\cap \Sigma _E$ is the unique component of $\Sigma _E$ that is included in G by Lemma 2 and Corollary 1. If C contains some point outside of $\overline{{\text {co}}}\,E$, then C might include more than one component of $\mathcal {G}_E$. In either case, each such component G along with $S_G=G\cap \Sigma _E$ (see Corollary 1) is unbounded by virtue of Lemma 3. More specifically, assertion (a) is correct by the proof of the implication (iii) $\Rightarrow$ (ii) of Lemma 3 and Proposition 2. Finally, (b) is true owing to Lemma 2 and Lemma 3. $\square$