Strong CHIP for Infinite System of Closed Convex Sets in Normed Linear Spaces ^{∗}

Chong Li Department of Mathematics Zhejiang University Hangzhou 310027, P. R. China Email: cli@zju.edu.cn

K. F. Ng Department of Mathematics Chinese University of Hong Kong Hong Kong, P. R. China Email: kfng@math.cuhk.edu.hk

Abstract. For a general (possibly infinite) system of closed convex sets in a normed linear space we provide several sufficient conditions for ensuring the strong conical hull intersection property. One set of sufficient conditions is given in terms of the finite subsystems while the other sets are in terms of the relaxed interiorpoint conditions together with appropriate continuity of the associated set-valued function on the (topologized) index set I. In the special case when I is finite and X is finite dimensional, one of these results reduces to a classical result of Rockafellar.

Key words: System of closed convex sets, interior-point condition, strong conical hull intersection property.

AMS(MOS) Subject Classifications. Primary, 90C34; 90C25 Secondary, 52A05; 41A29

1 Introduction

The notion of the strong CHIP (the strong conical hull intersection property) was introduced by Deutsch, Li and Ward in [13, 14] for a finite family of closed convex sets in an Euclidean space (or a Hilbert space) and has been successfully applied in the reformulation of some best approximation problems. This notion closely relates other fundamental concepts such as bounded linear regularity, G-property of Jameson, error bounds in convex optimization [1, 3] and the BCQ (the basic constraint qualification) as well as the perturbations for finite convex systems of inequalities. See [6, 7, 9, 13, 14, 15, 19, 20, 25, 26] and references therein, especially in [21] where the strong CHIP was defined for an arbitrary family of closed convex sets in a Banach space, and was utilized in the study of general systems of infinite convex inequalities such as the system that naturally arises from the problem of best restricted range approximation in the space C(Q) of complex-valued continuous functions on a compact metric space Q under quite general constraints. This problem was first presented and formulated by Smirnov and Smirnov in [33, 34] where each Ω_{t} was assumed to be a disk in ℂ. Later in [35, 36, 37] and also more recently in [18, 21], the constraint sets Ω_{t} have been relaxed but still remained to assume the strong interior-point condition (in particular, int Ω_{t} ≠ ∅ for each t ∈ Q). This unfortunately excludes the interesting case when some Ω_{t} is a line-segment or a singleton in ℂ. As demonstrated in an accompanying paper [23], the results obtained in the present paper have enabled us to study the restricted range approximation problem under much less restrictive assumptions by allowing the case that int Ω_{t} = ∅ for some t ∈ Q. The present paper is devoted to provide sufficient conditions for a (finite or infinite) family {C, C_{i} : i ∈ I} of closed convex sets in a Banach (or normed linear) space to have the strong CHIP.

∗The first author was supported in part by the National Natural Science Foundation of China (grant 10271025) and Program for New Century Excellent Talents in University. The second author was supported by a direct grant (CUHK)

and an Earmarked Grant from the Research Grant Council of Hong Kong.

In expanding and improving the known results on the sufficient conditions for strong CHIP for {C, C_{i} : i ∈ I} from the case when the index-set I is finite to the case when I may be infinite, this paper presents two types of results. One type is on the natural approach to answer the question: whether or not the following implication is valid:

{C, C_{j} : j ∈ J} has the strong CHIP at x_{0} for each finite subset J of I ⟹ {C, C_{i} : i ∈ I} has the strong CHIP at x_{0}.

While the answer of this question is negative in general (seeing [14, Example 1]), we provide some reasonable conditions in Section 5 to ensure the validity for the above implication. Another type of sufficient conditions presented in this paper is given more directly (in terms of the system itself rather than via its finite subsystems). In this connection, the starting point of our study is the following Theorem DLW. “DLW” refers to the authors Deutsch, Li and Ward of [13, 14] where the assertions regarding the sufficiency for (a) and for (b) were stated and proved in the Hilbert space setting but the arguments can be modified to suit the Banach space setting. For the sake of completeness and also for more convenient applications, we will present a direct proof for a slightly more general form in the next section (see also [27] for another approach).

Theorem DLW. Let I be a finite index set and {C, C_{i} : i ∈ I} be a finite family of nonempty closed convex sets in a Banach space X. Let x_{0} ∈ C ∩ (∩_{i∈I}C_{i}). Then the family {C, C_{i} : i ∈ I} has the strong CHIP at x_{0} provided that at least one of the following conditions is satisfied:

(a) C ⋂ (int ⋂_{i∈I} C_{i}) ≠ ∅.

(b) ri C ⋂ (⋂_{i∈I} C_{i}) ≠ ∅ and each C_{i} is a polyhedron.

(c) There exists a subset I_{0} of I such that C_{i} is a polyhedron for each i ∈ I \ I_{0} and

ri C ⋂ (int ⋂_{i∈I_{0}} C_{i}) ⋂ ( ⋂_{i∈I\I_{0}} C_{i}) ≠ ∅.  (1.1)

The sufficiency of (c) follows directly from (a) and (b). The condition (a) is sometimes referred to as the strong interior-point condition (or Slater condition, see, e.g, [14]) which is equivalent (as I is finite) to the following interior-point condition:

(a′) C ⋂ (⋂_{i∈I} int C_{i}) ≠ ∅.

As shown in [21], when I is infinite, the above (a), (b) and (c) are no longer sufficient for the strong CHIP. A natural condition that one would like to impose is the continuity assumption for the set-valued mapping i ↦ C_{i}; thus it is judicious for us to assume henceforth that

• the set I is a compact metric space.  (1.2)

(When I is finite, it will be regarded as a compact metric space under the discrete metric; needless to say that, in this case, the continuity assumption is automatically satisfied.)

Under an appropriate continuity assumption we show in Theorem 4.1 that (a) implies the strong CHIP at x_{0} ∈ C ∩ (∩_{i∈I}C_{i}) provided that C is finite dimensional or the set I_{C}^{rb}(x_{0}) of “C-relative boundary indices” for x_{0} is finite. We remark that even in the case when C is finite dimensional, our results are genuinely an extension of Theorem DLW as some (or all) sets C_{i} can be infinite dimensional. In a similar fashion other parts of Theorem DLW are extended in Section 4. In fact, we use the following condition, somewhat weaker than (c), to establish a sufficient condition result in Theorem 4.3. The family {C, C_{i} : i ∈ I} is said to satisfy the weak-strong interior-point condition with the pair (I_{1}, I_{2}) if there exist disjoint finite subsets I_{1}, I_{2} of I satisfying the following two properties:

• ri C ⋂ (int ⋂_{i∈I\(I_{1}∪I_{2})} C_{i}) ⋂ ( ⋂_{i∈I_{1}} ri C_{i}) ⋂_{i∈I_{2}} C_{i} ≠ ∅;  (1.3)

• C_{i} is a polyhedron for each i ∈ I_{2}.  (1.4)

This condition, in contrast to the interior-point condition, enables us to consider the case when some C_{i} is neither a polyhedron nor has an interior point. Specializing to the special case when I = I_{1} ∪ I_{2} (thus int(∩_{i∈I\(I_{1}∪I_{2})}C_{i}) to be read as X by convention), a corollary of Theorem 4.3 is the following infinite dimension extension of a result of Rockafellar [28, Corollary 23.8.1, p.223]:

Let I = J ∪ K be finite such that C_{k} is a polyhedron for each k ∈ K and suppose that

ri C ⋂ ( ⋂_{j∈J} ri C_{j}) ⋂ ( ⋂_{k∈K} C_{k}) ≠ ∅.  (1.5)

Then the system {C, C_{i} : i ∈ I} has the strong CHIP if at least one of the following conditions is satisfied.

(a) At least one of {C, C_{j} : j ∈ J} is finite dimensional.

(b) C_{j} is finite codimensional for each j ∈ J.

2 Notations and preliminary results

The notations used in the present paper are standard (cf. [8, 17]). In particular, we assume that X is a normed linear space throughout the whole paper, unless we state explicitly otherwise. We use B(x, ϵ) to denote the closed ball with center x and radius ϵ. For a set Z in X (or in ℝ^{n}), the interior (resp. relative interior, closure, convex hull, convex cone hull, linear hull, affine hull, boundary, relative boundary) of Z is denoted by int Z (resp. ri Z, \overline{Z}, conv Z, cone Z, span Z, aff Z, bd Z, rb Z), and the negative polar cone Z^{⊖} is the set defined by

Z^{⊖} = {x^{∗} ∈ X^{∗} : ⟨x^{∗}, z⟩ ≤ 0 for all z ∈ Z}.

The normal cone of Z at z_{0} is denoted by N_{Z}(z_{0}) and defined by N_{Z}(z_{0}) = (Z −z_{0})^{⊖}. For convenience of printing we sometimes use N(z_{0}; Z) in place of N_{Z}(z_{0}). Let A be a closed convex nonempty subset of X. The interior and boundary of Z relative to A are respectively denoted by rint_{A} Z and bd_{A} Z; they are defined to be respectively the interior and boundary of the set aff A ∩ Z in the metric space aff A. Thus, a point z ∈ rint_{A} Z if and only if there exists ε > 0 such that

z ∈ (aff A) ∩ B(z, ε) ⊆ Z,  (2.1)

while z ∈ bd_{A} Z if and only if z ∈ aff A and, for any ε > 0, (aff A) ∩ B(z, ε) intersects Z and its complement. Let ℝ_{−} denote the subset of ℝ consisting of all non-positive real numbers. For a proper extended real-valued convex function on X, the subdifferential of f at x ∈ X is denoted by ∂f(x) and defined by

∂f(x) = {z^{∗} ∈ X^{∗} : f(x) + Re ⟨z^{∗}, y − x⟩ ≤ f(y) for all y ∈ X},

where ⟨z^{∗}, x⟩ denotes the value of a functional z^{∗} in X^{∗} at x ∈ Z, i.e., ⟨z^{∗}, x⟩ = z^{∗}(x).

For simplicity of notations, we will usually assume (with the exception of Proposition 2.1 and Sections 5) that the scalar field of X is ℝ and that Re ⟨x^{∗}, x⟩ is to be replaced by ⟨x^{∗}, x⟩.

Remark 2.1. (a) Let f be a continuous convex function on X and x ∈ X with f(x) = 0. It is easy to see that cone(∂f(x)) ⊆ N_{f^{−1}(ℝ_{−})}(x), and that the equality holds if f is an affine function or if x is not a minimizer of f, see [8, Corollary 1, p. 56].

(b) The directional derivative of the function f at x in the direction d is denoted by f′_{+}(x, d):

f′_{+}(x, d) := lim_{t→0+} \frac{f(x + td) − f(x)}{t}.  (2.2)

We recall [8, Proposition 2.2.7] that, if x is a continuity point of f,

∂f(x) = {z^{∗} ∈ X^{∗} : ⟨z^{∗}, d⟩ ≤ f′_{+}(x, d) for all d ∈ X}  (2.3)

and

f′_{+}(x, d) = max{⟨z^{∗}, d⟩ : z^{∗} ∈ ∂f(x)}.  (2.4)

Let {A_{i} : i ∈ J} be a family of subsets of X. The set ∑_{i∈J} A_{i} is defined by

∑_{i∈J} A_{i} = \{ \begin{matrix}\{∑_{i∈J_{0}} a_{i} : a_{i} ∈ A_{i}, J_{0} ⊆ J being finite\} , & if J ≠ ∅, \\ \{0\}, & if J = ∅. \end{matrix}  (2.5)

The following concept of the strong CHIP plays an important role in optimization theory (see [1, 3, 9, 11, 12, 32])and is due to [13, 14] in the case when I is finite and [21] in the case when I is infinite.

Definition 2.1. Let {C_{i} : i ∈ I} be a collection of convex subsets of X and x ∈ ⋂_{i∈I} C_{i}. The collection is said to have (a) the strong CHIP at x if N_{⋂_{i∈I} C_{i}}(x) = ∑_{i∈I} N_{C_{i}}(x), that is

(⋂_{i∈I} C_{i} − x)^{⊖} = ∑_{i∈I}(C_{i} − x)^{⊖}.  (2.6)

(b) the strong CHIP if it has the strong CHIP at each point of ∩_{i∈I}C_{i}.

Consider a convex inequality system (CIS) defined by

g_{i}(x) ≤ 0, i ∈ I,  (2.7)

where x ∈ X and each g_{i} is a real continuous convex function on X. We always assume that the solution set S of the system (CIS) is nonempty, i.e.,

S := {x ∈ X : g_{i}(x) ≤ 0 for all i ∈ I} ≠ ∅.  (2.8)

Let G(·) denote the sup-function [17] of {g_{i}}:

G(x) := sup_{i∈I} g_{i}(x) for all x ∈ X.

Then S is also the solution set of the convex inequality:

G(x) ≤ 0.  (2.9)

In this paper we assume throughout that

G(x) < +∞ for all x ∈ X  (2.10)

and that G is continuous on X. These blanket assumptions are automatically satisfied if {g_{i} : i ∈ I} is locally uniformly bounded. Moreover, the continuity of G automatically follows from (2.10) if X is finite dimensional.

Let I(x) denote the set of all active indices i : I(x) = {i ∈ I : g_{i}(x) = G(x)}. Following [16, 24], we define

D′(x) := conv ∑_{i∈I(x)} ∂g_{i}(x), x ∈ X.  (2.11)

Note that, by (2.5), D′(x) = {0} if I(x) = ∅.

The following theorem will play a key role in Section 4. It is a known result, see for example [17, 24] in the special case when X is finite dimensional, and [22] for the general case (the proof presented in [22] is valid for normed linear spaces though the result was stated in the Banach space setting).

Theorem 2.1. Suppose that I is a compact metric space and that the function: i ↦ g_{i}(x) is upper semi-continuous for each x ∈ X. Let x_{0} ∈ C. Then I(x_{0}) ≠ ∅ and the following assertions hold.

(i) If span C is finite dimensional then

N_{C}(x_{0}) + ∂G(x_{0}) = N_{C}(x_{0}) + D′(x_{0}).  (2.12)

(ii) ∂G(x_{0}) = D′(x_{0}) provided that I(x_{0}) is finite.

Theorem 2.2 below is a slight extension (applicable to convex but, not necessarily closed, sets in a normed space). To prepare for the proof we begin with a simple lemma.

Lemma 2.1. Assume that C is a polyhedron in X defined by

C = ⋂_{i=1}^{k}{x ∈ X : ⟨h_{i}, x⟩ ≤ d_{i}},  (2.13)

where h_{i} ∈ X^{∗} \ {0} and d_{i} is a real number for each i = 1, …, k. Let x_{0} ∈ bd C and let I(x_{0}) = {i ∈ {1, …, k} : ⟨h_{i}, x_{0}⟩ = d_{i}}. Then

N_{C}(x_{0}) = cone {h_{i} : i ∈ I(x_{0})}.  (2.14)

Consequently, {C_{i} : i = 1, 2, …, n} has the strong CHIP if each C_{i} is a polyhedron of X.

Proof. We need only prove that the set on the left-hand side of (2.14) is contained in that on the right-hand side. To do this, suppose on the contrary that y^{∗} ∈ N_{C}(x_{0}) \ cone {h_{i} : i ∈ I(x_{0})}. Since cone {h_{i} : i ∈ I(x_{0})} is closed as I is finite, by the Separation Theorem, there exists an element x^{∗∗} ∈ X^{∗∗} such that

⟨x^{∗∗}, y^{∗}⟩ > 0 ≥ sup{⟨x^{∗∗}, h⟩ : h ∈ cone {h_{i} : i ∈ I(x_{0})}}.  (2.15)

Moreover, as I(x_{0}) is a finite set, there exists x ∈ X such that

⟨y^{∗}, x⟩ = ⟨x^{∗∗}, y^{∗}⟩ and ⟨h_{i}, x⟩ = ⟨x^{∗∗}, h_{i}, ⟩ for each i ∈ I(x_{0}).  (2.16)

Hence, by (2.15) and (2.16), we have that tx + x_{0} ∈ C for some t > 0 small enough. But ⟨y^{∗}, (tx + x_{0}) − x_{0}⟩ = t⟨y^{∗}, x⟩ > 0, which contradicts that y^{∗} ∈ N_{C}(x_{0}), and the lemma is proved.

Let Y be a subspace of X. We use N^{Y} to represent the normal cone operator taken in Y, namely for any subset A of Y, N_{A}^{Y} (x) is the set

N_{A}^{Y} (x) = {y^{∗} ∈ Y^{∗} : ⟨y^{∗}, z − x⟩ ≤ 0 for all z ∈ A}.  (2.17)

Corollary 2.1. Let Z ⊆ X be a closed subspace and C ⊆ X a polyhedron. Let x_{0} ∈ C ∩ Z and let N_{C∩Z}^{Z}(x_{0}) denote the normal cone of C ∩ Z at x_{0} taken in Z. Then, for each x_{0}^{∗} ∈ N_{C∩Z}^{Z}(x_{0}), there exists x^{∗} ∈ N_{C}(x_{0}) such that x^{∗} is an extension of x_{0}^{∗} (an extension of x_{0}^{∗} obtained from Lemma 2.1 will be referred to as a natural extension of x_{0}^{∗} ).

Theorem 2.2. Let I be a finite index set and {C, C_{i} : i ∈ I} be a finite family of nonempty convex sets in a nomed linear space X. Let x_{0} ∈ C ∩ (∩_{i∈I}C_{i}). Then the family {C, C_{i} : i ∈ I} has the strong CHIP at x_{0} provided that at least one of the conditions (a), (b) and (c) in Theorem DLW is satisfied.

Proof. In the case (a), if X is a Banach space and {C, C_{i} : i ∈ I} is a family of nonempty closed convex sets, the proof is the same as that given in [14] except that, here we apply [2, Theorem 2.6, p189] instead of [2, Corollary 2.5, p113]. Note further that the result is valid for any normed linear space X and any family {C, C_{i} : i ∈ I} of nonempty convex sets. However this observation does not constitute a genuine extension. Indeed, let U denote the open unit ball of \overline{X}. Let \overline{C} and \overline{C}_{i} respectively denote the closures of C and C_{i} in \overline{X}. By (a), there exists c ∈ C and ϵ > 0 such that

(c + ϵU) ∩ X ⊆ C_{i} for each i ∈ I.  (2.18)

We claim that

c + ϵU ⊆ \overline{C}_{i} for each i ∈ I.  (2.19)

Indeed, let u ∈ U. Then there exists a sequence {x_{n}} ⊆ U ∩ X convergent to u. Then by (2.18) \{c + ϵx_{n}\} ⊆ C_{i}, which implies that c + ϵu ∈ \overline{C}_{i}. Thus (2.19) is clear. Let x_{0} ∈ C ⋂(⋂_{i∈I} C_{i}). Then one can apply the Banach space version of (a) in Theorem 2.2 to conclude that

N_{\overline{C}∩(∩_{i∈I} \overline{C}_{i})}(x_{0}) ⊆ N\overline{C}(x_{0}) + ∑_{i∈I} N_{\overline{C}_{i}}(x_{0})

and hence

N_{C∩(∩_{i∈I}C_{i})}(x_{0}) = N\overline{C∩(∩_{i∈I}C_{i})}(x_{0}) ⊆ N_{C}(x_{0}) + ∑_{i∈I} N_{C_{i}}(x_{0})

because \overline{C ∩ (∩_{i∈I}C_{i})} = \overline{C} ∩ (∩_{i∈I} \overline{C}_{i}) thanks to (2.19). This completes the proof of (a).

Now let us verify the conclusion in the case (b). Let x_{0} ∈ C ⋂(⋂_{i∈I} C_{i}). and let Z denote the subspace spanned by C − x_{0}. Since ri C ⋂ (⋂_{i∈I} C_{i}) ≠ ∅, the intersection of the interior of C − x_{0} in the subspace Z and the set ⋂_{i∈I}(C_{i} − x_{0}) is nonempty. By the case (a) and Lemma 2.1 (applied to Z in place of X), we obtain that

N_{(C−x_{0})∩(∩_{i∈I}(C_{i}−x_{0}))}^{Z}(0) = N_{C−x_{0}}^{Z}(0)+N_{Z∩(∩_{i∈I}(C_{i}−x_{0}))}^{Z}(0) = N_{C−x_{0}}^{Z}(0)+∑_{i∈I} N_{Z∩(C_{i}−x_{0})}^{Z}(0). (2.20)

Let x^{∗} ∈ N_{C∩(∩C_{i})}(x_{0}). Then x^{∗}|_{Z} ∈ N_{(C−x_{0})∩(∩_{i∈I}(C_{i}−x_{0}))}^{Z}(0); consequently, by (2.20), there exist x_{0}^{∗} ∈ N_{C−x_{0}}^{Z}(0) and x_{i}^{∗} ∈ N_{Z∩(C_{i}−x_{0})}^{Z}(0) for each i ∈ I such that

x^{∗}|_{Z} = x_{0}^{∗} + ∑_{i∈I} x_{i}^{∗} on Z  (2.21)

Let x_{0}^{∗} ∈ X^{∗} be an extension of x_{0}^{∗}. Then, as C − x_{0} ⊆ Z, x_{0}^{∗} ∈ N_{C−x_{0}}(0) = N_{C}(x_{0}). Also, for each i ∈ I, as C_{i} − x_{0} is a polyhedron in X, there exists a natural extension x_{i}^{∗} ∈ N_{C_{i}−x_{0}}(0) of x_{i}^{∗} by Corollary 2.1. and hence x_{i}^{∗} ∈ N_{C_{i}}(x_{0}) with x_{i}^{∗}|_{Z} = x_{i}^{∗}. for each i ∈ I. Do this for each i ∈ I and let y^{∗} = x^{∗} − x_{0}^{∗} − ∑_{i∈I} x_{i}^{∗}. Then y^{∗} ∈ N_{C}(x_{0}) by (2.21). Hence, x^{∗} = y^{∗} + x_{0}^{∗} + ∑_{i∈I} x_{i}^{∗} ∈ N_{C}(x_{0}) + ∑_{i∈I} N_{C_{i}}(x_{0}) and the conclusion in the case (b) is proved. Therefore the proof of Theorem 2.2 is complete as (c) follows from (a) and (b).

For a closed convex subset W of X, let P_{W} denote the projection operator defined by

P_{W} (x) = {y ∈ W : ∥x − y∥ = d_{W} (x)},

where d_{W} (x) denotes the distance from x to W. Recall that the duality map J from X to 2^{X^{∗}} is defined by

J(x) := {x^{∗} ∈ X^{∗} : ⟨x^{∗}, x⟩ = ∥x∥^{2}, ∥x^{∗}∥ = ∥x∥}.  (2.22)

In fact, J(x) = ∂ϕ(x), where ϕ(x) := \frac{1}{2}∥x∥^{2}. Thus a Banach space X is smooth if and only if for each x ∈ X the duality map is single-valued. We also need the following proposition which was established independently by Deutsch [10] and Rubenstein [29] (see also [4]).

Proposition 2.1. Let W be a convex set in X. Then for any x ∈ X, z_{0} ∈ P_{W} (x) if and only if z_{0} ∈ W and there exists x^{∗} ∈ J(x − z_{0}) such that Re ⟨x^{∗}, z − z_{0}⟩ ≤ 0 for any z ∈ W, that is, J(x − z_{0}) ∩ N_{W} (z_{0}) ≠ ∅. In particular, when X is smooth, z_{0} ∈ P_{W} (x) if and only if z_{0} ∈ W and J(x − z_{0}) ∈ N_{W} (z_{0}).

3 Extended Minkowski functional, interior-point condition and continuity condition

Recall that I denotes an index-set which is assumed to be a compact metric space. For convenience, a family {C, C_{i} : i ∈ I} is called a closed convex set system with base-set C (CCS-system with base-set C) if C and C_{i} are nonempty closed convex subsets of X for each i ∈ I.

Definition 3.1. A CCS-system {C, C_{i} : i ∈ I} with base-set C is said to satisfy:

(i) the C-relative interior-point condition if

C ⋂ (⋂_{i∈I} rint_{C}C_{i}) ≠ ∅;  (3.1)

(ii) the strong C-relative interior-point condition if

C ⋂ (rint_{C} ⋂_{i∈I} C_{i}) ≠ ∅;  (3.2)

(iii) the weak-strong C-relative interior-point condition with the pair (I_{1}, I_{2}) if there exist two disjoint finite subsets I_{1} and I_{2} of I such that each C_{i} (i ∈ I_{2}) is a polyhedron and

ri C ⋂ (rint_{C} ⋂_{i∈I\(I_{1}∪I_{2})} C_{i}) ⋂ ( ⋂_{i∈I_{1}} ri C_{i}) ⋂_{i∈I_{2}} C_{i} ≠ ∅.  (3.3)

(iv) the interior-point condition (resp. the strong interior-point condition, the weak-strong interiorpoint condition with the pair (I_{1}, I_{2})) if the operation “rint_{C}” in (3.1) (resp. (3.2), (3.3)) is replaced with “int”.

(v) any point x̄ belonging to the set on the left-hand side of (3.1) (resp. (3.2), (3.3)) is called a Crelative interior point (resp. a strong C-relative interior point, a weak-strong C-relative interior point with the pair (I_{1}, I_{2})) of the CCS-system {C, C_{i} : i ∈ I}. Similarly, the notion of an interior point (resp. a strong interior point, a weak-strong interior point with the pair (I_{1}, I_{2})) of the CCS-system {C, C_{i} : i ∈ I} is defined.

It is trivial that (3.2)⟹(3.1). The converse also holds in some cases, one of which will be described in terms of continuity of some set-valued functions. For set-valued functions there are many different notions of continuity. In Definitions 3.2 and 3.3 below, we recall two frequently used ones. We assume that Q is a compact metric space.

Definition 3.2. Let Y be a normed linear space. Then the set-valued function F : Q → 2^{Y} \ {∅} is said to be

(i) lower semicontinuous at t_{0} ∈ Q, if, for any y_{0} ∈ F (t_{0}) and any ϵ > 0, there exists an open neighbourhood U(t_{0}) of t_{0} such that, for each t ∈ U(t_{0}), B(y_{0}, ϵ) ∩ F (t) ≠ ∅.

(ii) locally uniform lower semicontinuous at t_{0} ∈ Q, if, for any y_{0} ∈ F (t_{0}), there exists an open neighborhood V (y_{0}) of y_{0} such that for any ϵ > 0, there exists an open neighbourhood U(t_{0}) of t_{0} such that B(y, ϵ) ∩ F (t) ≠ ∅ holds for each t ∈ U(t_{0}) and each y ∈ V (y_{0}) ∩ F (t_{0}).

(iii) upper semicontinuous at t_{0} ∈ Q if, for any open neighbourhood V of F (t_{0}), there exists an open neighbourhood U(t_{0}) of t_{0} such that F (t) ⊆ V for each t ∈ U(t_{0}).

(iv) lower semicontinuous (resp. locally uniform lower semicontinuous, upper semicontinuous) on Q if it is lower semicontinuous (resp. locally uniform lower semicontinuous, upper semicontinuous) at each t ∈ Q.

Definition 3.3. (cf.[31, p.55]) Let F : Q → 2^{Y} be a set-valued function defined on Q and let t_{0} ∈ Q. Then F is said to be

(i) upper Kuratowski semicontinuous at t_{0} if, for any sequence {t_{k}} ⊆ Q, the relations lim_{k→∞} t_{k} = t_{0}, lim_{k→∞} x_{t_{k}} = x_{t_{0}}, x_{k} ∈ F (t_{k}), k = 1, 2, … imply x_{t_{0}} ∈ F (t_{0}).

(ii) lower Kuratowski semicontinuous at t_{0} if, for any sequence {t_{k}} ⊆ Q, the relations lim_{k→∞} t_{k} = t_{0}, y_{0} ∈ F (t_{0}) imply lim_{k→∞} d_{F (t_{k})}(y_{0}) = 0;

(iii) Kuratowski continuous at t_{0} if F is both upper Kuratowski semicontinuous and lower Kuratowski semicontinuous at t_{0}.

(iv) Kuratowski continuous on Q if it is Kuratowski continuous at each point of Q.

Remark 3.1. Clearly,

(i) F is upper semicontinuous ⟹F is upper Kuratowski semicontinuous. (ii) F is lower semicontinuous ⟺F is lower Kuratowski semicontinuous.

Moreover, the converse of (i) holds provided that the union set ∪_{t∈Q}F (t) is compact.

The following two propositions provide some useful reformulations regarding various lower semicontinuities. Since the proofs are similar, we shall only prove the first proposition.

Proposition 3.1. Let F : Q → 2^{Y} \ {∅} be a set-valued function. Let t_{0} ∈ Q. Then the following statements are equivalent.

(i) F is lower semicontinuous at t_{0}. (ii) For any y_{0} ∈ F (t_{0}), there exists y_{t} ∈ F (t) for each t ∈ Q such that lim_{t→t_{0}} ∥y_{t} − y_{0}∥ = 0. (iii) For any y_{0} ∈ F (t_{0}), lim_{t→t_{0}} d_{F (t)}(y_{0}) = 0.

Proof. (i)⟹(ii). Let y_{0} ∈ F (t_{0}). Then, by (i), for each positive k, there exists an open neighbourhood U_{k}(t_{0}) of t_{0} such that

B(y_{0}, \frac{1}{k}) ∩ F (t) ≠ ∅ for each t ∈ U_{k}(t_{0}).  (3.4)

Without loss of generality, we may assume that U_{k+1}(t_{0}) ⊆ U_{k}(t_{0}) for each k and ⋂_{k≥1} U_{k}(t_{0}) = {t_{0}} because Q is a metric space. Now we construct y_{t} ∈ F (t) for each t ∈ Q as follows.

\begin{matrix}y_{t} ∈ F (t) & if t ∈ Q \ U_{1}(t_{0}), \\ y_{t} ∈ B(y_{0}, \frac{1}{k}) ∩ F (t) & if t ∈ U_{k}(t_{0}) \ U_{k+1}(t_{0}) k = 1, …, \\ y_{0} & if t = t_{0}. \end{matrix}  (3.5)

Then lim_{t→t_{0}} y_{t} = y_{0}.

(ii)⟹(iii). It is trivial.

(iii)⟹(i). Let y_{0} ∈ F (x_{0}) and ϵ > 0. By (iii), there exists an open neighbourhood U(t_{0}) of t_{0} such that, for each t ∈ U(t_{0}), one has that d_{F (t)}(y_{0}) < ϵ and so B(y_{0}, ϵ) ∩ F (t) ≠ ∅. Therefore (i) holds. The proof is complete.

The following proposition can be proved similarly.

Proposition 3.2. Let F : Q → 2^{Y} \ {∅} be a set-valued function. Let t_{0} ∈ Q. Then the following statements are equivalent.

(i) F is locally uniform lower semicontinuous at t_{0}.

(ii) for any y_{0} ∈ F (t_{0}), there exists an open neighborhood V (y_{0}) of y_{0} such that for any y ∈ V (y_{0}) ∩ F (t_{0}), there exists z_{t}(y) ∈ F (t) for each t ∈ Q such that lim_{t→t_{0}} ∥z_{t}(y) − y∥ = 0 holds uniformly on V (y_{0}) ∩ F (t_{0}).

(iii) for any y_{0} ∈ F (t_{0}), there exists an open neighborhood V (y_{0}) of y_{0} such that

lim_{t→t_{0}} sup_{y∈V (y_{0})∩F (t_{0})} d_{F (t)}(y) = 0.

(iv) for any y_{0} ∈ F (t_{0}), there exists a neighborhood V (y_{0}) of y_{0} such that, for any ϵ > 0, there exists a neighborhood U(t_{0}) of t_{0} satisfying

V (y_{0}) ∩ F (t_{0}) ⊆ ⋂_{t∈U(t_{0})} F (t)^{ε},

where A^{ε} is defined by

A^{ε} = {y ∈ Y : d_{A}(y) < ε}.

Recalling our blanket assumption (1.2), and the definition of CCS-systems made at the beginning of this section, we state our first main result of this section.

Theorem 3.1. Let {C, C_{i} : i ∈ I} be a CCS-system with base-set C, and Let x̄ ∈ C. Suppose that the set-valued function i ↦ (aff C) ∩ C_{i} is locally uniform lower semicontinuous on I. Then x̄ is a C-relative interior point of the system if and only if it is a strong C-relative interior point of the system.

Proof. We need only to prove the necessity part. Assume without loss of generality that

0 = x̄ ∈ C ⋂ (⋂_{i∈I} rint_{C}C_{i}) .  (3.6)

Then Z := aff C is a vector subspace of X. It suffices to show that

0 ∈ C ⋂ (rint_{C} ⋂_{i∈I} C_{i}) .  (3.7)

Clearly, we need only to show that

inf_{i∈I} d_{bd_{Z}C_{i}}(0) > 0.  (3.8)

(Indeed if d_{bd_{Z}C_{i}}(0) > γ then Z ∩ B(0, γ) ⊆ C_{i}.) Suppose on the contrary that (3.8) does not hold. Then, by the compactness of I, there exist a convergent sequence (i_{n}) ⊆ I (say with limit i_{0} ∈ I) and a sequence (y_{i_{n}}) with y_{i_{n}} ∈ bd_{Z} C_{i_{n}} for each n such that lim_{n} ∥y_{i_{n}}∥ = 0. Write

Ĉ_{i} = Z ∩ C_{i}, i ∈ I.  (3.9)

By assumptions, i ↦ Ĉ_{i} is locally uniform lower semicontinuous at i_{0}. By (iv)of Proposition 3.2 (applied to 0, i_{0}, in place of y_{0}, t_{0}), there exists a δ ∈ (0, 1) such that for any ε > 0 there exists a neighborhood U_{ε}(i_{0}) of i_{0} such that

B(0, δ) ∩ Ĉ_{i_{0}} ⊆ ⋂_{i∈U_{ε}(i_{0})} Ĉ_{i}^{ε}.  (3.10)

In view of (3.6), we may assume in addition that

B(0, δ) ∩ Z ⊆ Ĉ_{i_{0}}  (3.11)

(take a smaller δ > 0 if necessary). Combining the above two inclusions, we have

B(0, δ) ∩ Z ⊆ ⋂_{i∈U_{ε}(i_{0})} Ĉ_{i}^{ε}.  (3.12)

Let us fix an ε ∈ (0, \frac{δ}{3}) and take α > 0 such that \frac{3}{2}ε < α < δ/2 (hence α < \frac{1}{2} as δ < 1). We fix a natural number n which is large enough so that

∥y_{i_{n}}∥ < \frac{δ}{2} and i_{n} ∈ U_{ε}(i_{0}).  (3.13)

For simplicity of notations, we henceforth write i for the i_{n} with the above n. Since y_{i} is a (relative) boundary point of C_{i} ∩ Z = Ĉ_{i} in the vector subspace Z of X and since Ĉ_{i} has a nonempty relative interior (containing the origin) by (3.6), the Separation Theorem implies that NĈ_{i}(y_{i})|_{Z} ≠ {0}. Hence, by the Hahn-Banach Theorem, there exists x_{i}^{∗} ∈ NĈ_{i}(y_{i}) such that ∥x_{i}^{∗}|_{Z}∥ is of norm 1. Take x_{i}^{ε} ∈ Z such that ∥x_{i}^{ε}∥ = 1 and ⟨x_{i}^{∗}, x_{i}^{ε}⟩ ≥ 1 − \frac{ε}{2}. Define z_{i} := y_{i} + αx_{i}^{ε}. Then z_{i} ∈ Z,

∥(y_{i} + x_{i}^{ε}) − z_{i}∥ = ∥(1 − α)x_{i}^{ε}∥ = 1 − α

and it follows from the triangle inequality that, for any y ∈ Ĉ_{i},

∥z_{i} − y∥ ≥ ∥(y_{i} + x_{i}^{ε}) − y∥ − (1 − α)
≥ ⟨x_{i}^{∗}, y_{i} − y⟩ + ⟨x_{i}^{∗}, x_{i}^{ε}⟩ − (1 − α)
≥ ⟨x_{i}^{∗}, x_{i}^{ε}⟩ − (1 − α)
≥ (1 − \frac{ε}{2}) − (1 − α)
= α − \frac{ε}{2} > ε.

Therefore z_{i} ∉ Ĉ_{i}^{ε}. This contradicts (3.12) and (3.13) because

∥z_{i}∥ ≤ ∥z_{i} − y_{i}∥ + ∥y_{i}∥ = ∥αx_{i}^{ε}∥ + ∥y_{i}∥ < α + \frac{δ}{2} ≤ δ.

Thus (3.8) must hold and the proof is complete.

Let A and Ĉ be two closed convex subsets of X with 0 ∈ rintĈ A. In the following we will show that A admits a “Ĉ-extended Minkowski functional” p_{A} in the sense that p_{A} is a continuous sublinear functional on X such that its restriction p_{A}|_{aff Ĉ} equals the Minkowski functional of A ∩ aff Ĉ in the vector subspace aff Ĉ of X. Note that in this case one has, for each z ∈ aff Ĉ,

p_{A}(z) ≤ 1 ⟺ z ∈ A  (3.14)
p_{A}(z) = 1 ⟺ z ∈ bd_{C} A.  (3.15)

Lemma 3.1. Suppose that 0 ∈ rintĈ A:

0 ∈ B(0, α) ∩ aff Ĉ ⊆ A  (3.16)

for some α > 0, where A and Ĉ are closed convex subsets of X. Denote the closure of aff Ĉ(= span Ĉ) by Z and let Ã denote the closed convex hull of the set (A ∩ Z) ∪ (B(0, α)). Then

(i) Ã is a closed convex set in X with nonempty interior such that

Ã ∩ Z = A ∩ Z and 0 ∈ int Ã.  (3.17)

(ii) The corresponding Minkowski functional qÃ (in the usual sense) on X has the properties

qÃ(x) ≤ \frac{1}{α}∥x∥ for each x ∈ X  (3.18)

and

qÃ(x) = inf{λ ≥ 0 : x ∈ λ(A ∩ Z)} for each x ∈ Z,  (3.19)

(that is qÃ is a Ĉ-extended Minkowski functional of A).

Proof. Let D denote the convex hull of the set (A ∩ Z) ∪ (B(0, α)). Then Ã = \overline{D} and B(0, α) ⊆ D. Hence, by elementary functional analysis, the Minkowski functional of Ã coincides with that of D, and (3.18) holds. Hence, to prove (3.19) it suffices to show that

D ∩ Z = A ∩ Z.  (3.20)

Let x = λ_{1}a + λ_{2}b ∈ D ∩ Z with a ∈ A ∩ Z, b ∈ B(0, α) and λ_{1}, λ_{2} ∈ (0, 1) such that λ_{1} + λ_{2} = 1. Then b ∈ Z. We claim that b ∈ A. In fact, since b ∈ Z, there exists a sequence {b_{k}} ⊂ aff Ĉ such that b_{k} → b. If b ∈ int B(0, α), then b_{k} ∈ B(0, α) for all k large enough. This implies that b_{k} ∈ B(0, α) ∩ aff Ĉ for for all such k. Therefore, by (3.16), b_{k} ∈ A and hence b ∈ A. Thus assume that b ∈ bd B(0, α). Define b̃_{k} = \frac{αb_{k}}{∥b_{k}∥} for each k. Then b̃_{k} ∈ B(0, α) ∩ aff Ĉ and b̃_{k} → b for each k. Hence b ∈ A by (3.16). Therefore our claim stands and hence x ∈ A as A is convex. This shows (3.20), and hence (3.19) are true. To verify (3.17), let z ∈ Ã ∩ Z. Then qÃ(z) ≤ 1 and one can apply (3.19) to conclude that z ∈ A because A is closed and z ∈ Z. Thus (3.17) is seen to be true and the proof is complete.

Note. The set Ã will be referred to as a Ĉ-Minkowski extension of A (though it also depends on α in (3.16)).

For the remainder of this paper, {C, C_{i} : i ∈ I} denotes a CCS-system with base-set C as defined at the beginning of this section. Now we state the second main result of this section.

Theorem 3.2. Let x̄ ∈ C ∩ (∩_{i∈I}C_{i}) and let Ĉ := C − x̄, Ĉ_{i} := C_{i} − x̄ for each i ∈ I. Then the following statements are equivalent:

(i) x̄ is a strong C-relative interior point of the CCS-system {C, C_{i} : i ∈ I}, namely,

x̄ ∈ C ∩ rint_{C} (∩_{i∈I}C_{i}) .  (3.21)

(ii) For each i ∈ I, there exists a Ĉ-extended Minkowski functional pĈ_{i} of the set Ĉ_{i} such that the sup-function P (·) of {pĈ_{i}(·)} defined by

P (x) := sup_{i∈I} pĈ_{i}(x), x ∈ X  (3.22)

is continuous on X.

Moreover, if we add an additional assumption that the set-valued map i ↦ (aff C) ∩ C_{i} is lower semicontinuous then the above (ii) can be replaced by a stronger one:

˜(ii) (ii) holds and i ↦ pĈ_{i}(x) is upper semicontinuous for each x ∈ X.

Proof. (ii)⟹(i). Let Z := aff Ĉ = span Ĉ. By (3.14), Z ∩ Ĉ_{i} = {z ∈ Z : pĈ_{i}(z) ≤ 1} and hence

Z ⋂ (⋂_{i∈I} Ĉ_{i}) = {z ∈ Z : P (z) ≤ 1}.

By continuity assumption of P, it follows that 0 ∈ rint_{Z}(∩_{i∈I} Ĉ_{i}) and hence x̄ ∈ C ∩ rint_{C} (∩_{i∈I}C_{i}). Thus (3.21) is seen to hold.

(i)⟹(ii). By (3.21), there exists α > 0 such that

B(0, α) ∩ Z ⊆ Ĉ_{i} for each i ∈ I.  (3.23)

Then, by Lemma 3.1, there exists a Ĉ-extended Minkowski functional pĈ_{i} of Ĉ_{i} such that pĈ_{i} is the Minkowski functional of the closed convex hull of (A ∩ Z) ∪ (B(0, α)) and in particular

pĈ_{i}(x) ≤ \frac{1}{α}∥x∥ for each x ∈ X.  (3.24)

Hence, by definition of P,

P (x) ≤ \frac{1}{α}∥x∥ for each x ∈ X  (3.25)

and thus P is continuous by an elementary argument. This establishes the implication (i)⟹(ii).

For the remainder of the proof we assume, in addition, that the set-valued map i ↦ (aff C) ∩ C_{i} is lower semicontinuous and hence the set-valued map i ↦ Z ∩ Ĉ_{i} is lower semicontinuous. Then to prove (i) ⟹ ˜(ii), it remains to show for any i_{0} ∈ I and any x ∈ X that

lim sup_{i→i_{0}} pĈ_{i}(x) ≤ pĈ_{i_{0}} (x).  (3.26)

Suppose not. Then there exist i_{0} ∈ I and x ∈ X such that

lim sup_{i→i_{0}} pĈ_{i}(x) > 1 > pĈ_{i_{0}} (x).  (3.27)

Then x ∈ co ((Ĉ_{i_{0}} ∩ Z) ∪ (B(0, α))) and so x = λ_{1}b + λ_{2}z for some b ∈ B(0, α), z ∈ Ĉ_{i_{0}} ∩ Z and some λ_{1}, λ_{2} ∈ [0, 1] with λ_{1} + λ_{2} = 1. Since the set-valued function i ↦ Ĉ_{i} ∩ Z is lower semicontinuous at i_{0}, there exists z_{i} ∈ Ĉ_{i} ∩ Z for each i ∈ I such that z_{i} → z as i → i_{0}. Define x_{i} = λ_{1}b + λ_{2}z_{i} for each i ∈ I. Then x_{i} ∈ co ((Ĉ_{i} ∩ Z) ∪ (B(0, α))) and so pĈ_{i}(x_{i}) ≤ 1. Consequently it follows from (3.25) that

pĈ_{i}(x) ≤ pĈ_{i}(x − x_{i}) + pĈ_{i}(x_{i}) ≤ \frac{1}{α}∥x − x_{i}∥ + 1

and hence that lim sup_{i→i_{0}} pĈ_{i}(x) ≤ 1. This contradicts (3.27). Therefore (3.26) must hold for each i_{0} ∈ I and x ∈ X.

The following proposition deals with a special case in Theorem 3.2 by deleting the words “relative” and “Ĉ-extended” respectively from (i) and (ii). We will omit the proof as it is similar to that of Theorem 3.2.

Theorem 3.3. Let x̄ ∈ C ∩ (∩_{i∈I}C_{i}). Then the following statements are equivalent:

(i) x̄ is a strong interior point of the CCS-system {C, C_{i} : i ∈ I}, namely,

x̄ ∈ C ∩ int (∩_{i∈I}C_{i}) .

(ii) x̄ ∈ int C_{i} for each i ∈ I and the sup-function P (·) of {pĈ_{i}(·)} is continuous on X, where pĈ_{i} is the Minkowski functional of the set Ĉ_{i} := C − x̄.

Moreover, if we add an additional assumption that the set-valued map i ↦ C_{i} is lower semicontinuous then the above (ii) can be replaced by a stronger one:

˜(ii) (ii) holds and i ↦ pĈ_{i}(x) is upper semicontinuous for each x ∈ X.

4 interior-point condition and the strong CHIP

For the remainder of this paper, we assume that I is a compact metric space and that {C, C_{i} : i ∈ I} is a CCS-system with base-set C as in the beginning of the preceding section. Our main results are to provide sufficient conditions for ensuring the strong CHIP. For x_{0} ∈ C ∩ ( ∩_{i∈I} C_{i}), let I_{C}^{rb}(x_{0}) = {i ∈ I : x_{0} ∈ bd_{C} C_{i}}. Since bd_{C} C_{i} = bd C_{i} \ int_{C} C_{i},

I_{C}^{rb}(x_{0}) ⊆ {i ∈ I : x_{0} ∈ bd C_{i}}.  (4.1)

Theorem 4.1. Let x_{0} ∈ C ∩ ( ∩_{i∈I} C_{i}). Then the CCS-system {C, C_{i} : i ∈ I} has the strong CHIP at x_{0} if the following conditions are satisfied.

(a) The system {C, C_{i} : i ∈ I} satisfies the strong C-relative interior-point condition.

(b) The set-valued mapping i ↦ (aff C) ∩ C_{i} is lower semicontinuous on I.

(c) The pair {aff C, C_{i}} has the strong CHIP at x_{0} for each i ∈ I.

(d) C is either finite dimensional or I_{C}^{rb}(x_{0}) is a finite set. Moreover, the same conclusion also holds if (a), (b) and (c) are replaced simultaneously by (a^{∗}) and (b^{∗}), where

(a^{∗}) Same as (a) but delete the word “relative”.

(b^{∗}) The set-valued mapping i ↦ C_{i} is lower semicontinuous on I.

Remark 4.1. In view of Theorem 2.2, (a^{∗})⟹ (c) because (a^{∗}) implies int C_{i} ⊇ int (∩_{i∈I} C_{i})∩C ≠ ∅.

Proof of Theorem 4.1. By the assumptions (a) and (b), let x̄, Ĉ, Ĉ_{i}, pĈ_{i} and P be as in parts (i), (ĩi) of Theorem 3.2. In particular, P is continuous, the function i ↦ pĈ_{i}(x) is upper semicontinuous for each x ∈ X, and for each x ∈ aff C and i ∈ I it holds that

pĈ_{i}(x − x̄) ≤ 1 ⟺ x ∈ C_{i},  (4.2)
pĈ_{i}(x − x̄) = 1 ⟺ x ∈ bd_{C}C_{i},  (4.3)

where (4.3) holds by (3.15). [Note. The above considerations are valid by Proposition 3.5 if one assumes (a^{∗})+(b^{∗}) instead of (a)+(b); in fact, in this case, (4.2) and (4.3) hold for all x ∈ X (not only those x in aff C).]

Define, for each i ∈ I,

g_{i}(x) = pĈ_{i}(x − x̄) − 1 for each x ∈ X

and let G : X → ℝ be defined by G(x) := sup_{i∈I} g_{i}(x) for each x ∈ X. Then G is continuous and the function i ↦ g_{i}(x) is upper semicontinuous for each x ∈ X. Thus, one can apply Theorem 2.1 to conclude that

N_{C}(x_{0}) + ∂G(x_{0}) = N_{C}(x_{0}) + cone ∑_{i∈I(x_{0})} ∂ g_{i}(x_{0})  (4.4)

provided that the following condition (d′) is satisfied:

(d′) C is finite dimensional or I(x_{0}) is finite. To prove the theorem, we need only to prove the inclusion

N_{C∩(∩_{i∈I}C_{i})}(x_{0}) ⊆ N_{C}(x_{0}) + ∑_{i∈I} N_{C_{i}}(x_{0}).  (4.5)

as the reverse inclusion is evident. Note that, by (4.2),

N_{C ⋂(⋂_{i∈I} C_{i})}(x_{0}) = N_{C ⋂(⋂_{i∈I} g_{i}^{−1}(ℝ_{−}))}(x_{0}) = N_{C ⋂(G^{−1}(ℝ_{−}))}(x_{0}) = N_{C}(x_{0}) + N_{G^{−1}(ℝ_{−})}(x_{0}) (4.6)

thanks to Theorem DLW because x̄ ∈ C ∩ int G^{−1}(ℝ_{−}) as G(x̄) = −1. Thus (4.5) is seen to hold if N_{G^{−1}(ℝ_{−})}(x_{0}) = 0. Therefore we may henceforth assume that G(x_{0}) = 0. Then, referring the corresponding definition made in (2.11), I(x_{0}) = {i ∈ I : g_{i}(x_{0}) = 0}. Since x_{0} ∈ C it follows from (4.3) that

I(x_{0}) = I_{C}^{rb}(x_{0}).  (4.7)

Thus, by assumption (d), the condition (d′) holds. Recall from [8, Corollary 1, p. 56] that

N_{g_{i}^{−1}(ℝ_{−})}(x_{0}) = \{ \begin{matrix}cone ∂g_{i}(x_{0}), & i ∈ I(x_{0}), \\ 0, & i ∉ I(x_{0}) \end{matrix}  (4.8)

and similarly

N_{G^{−1}(ℝ_{−})}(x_{0}) = cone ∂G(x_{0}).  (4.9)

Hence, by (4.6) and (4.4), we have

N_{C ⋂(⋂_{i∈I} C_{i})}(x_{0}) = N_{C}(x_{0}) + cone ∂G(x_{0})
= N_{C}(x_{0}) + cone ∑_{i∈I(x_{0})} ∂ g_{i}(x_{0})
= N_{C}(x_{0}) + ∑_{i∈I(x_{0})} N_{g_{i}^{−1}(ℝ_{−})}(x_{0})
⊆ N_{C}(x_{0}) + ∑_{i∈I(x_{0})} N_{C_{i}∩aff C}(x_{0})

(4.10)

because g_{i}^{−1}(ℝ_{−}) ⊇ C_{i} ∩ aff C. This implies that (4.5) holds because for each i ∈ I(x_{0}) = I_{C}^{rb}(x_{0}), one has from the assumption (c) that

N_{C_{i}∩aff C}(x_{0}) = N_{C_{i}}(x_{0}) + N_{aff C}(x_{0})

(note also that N_{C}(x_{0}) = N_{C}(x_{0}) + N_{aff C}(x_{0}) as C ⊆ aff C). Thus the first part of the theorem is proved. The proof of the second part is almost the same because, if the assumptions (a)+(b)+(c) are replaced by (a^{∗}) + (b^{∗}), then (4.7) remains true (noting that x̄ ∈ C ∩ int C_{i} and x_{0} ∈ C ∩ C_{i} for each i ∈ I), and that N_{C_{i}∩aff C}(x_{0}) in (4.10) is to be replaced by N_{C_{i}}(x_{0}) .

Remark 4.2. As the assumption (c) is used only at the end of the proof, (4.10) is valid even if (c) is not assumed in Theorem 4.1.

Remark 4.3. The result in the first part of Theorem 4.1 is not true if the assumptions are replaced by (a), (c), (d), and (b^{∗}) (instead of (b)), as shown by the following example.

Example 4.1 Let X = ℝ^{2} and let I = \{0, 1, \frac{1}{2}, …\}. For each i ∈ I, let C_{i} be the closed convex subset of ℝ^{2} that is bounded by four line-segments l_{1}, l_{2}, l_{3}, l_{4} and the curve defined by

t_{2} = i(t_{1} + 1 + i)^{2} ∀t_{1} ∈ [−2, −1 − i],

where l_{1}, l_{2}, l_{3} and l_{4} are respectively defined by

\begin{matrix}l_{1} : & t_{1} = −2 ∀t_{2} ∈ [i(1 − i)^{2}, 1], \\ l_{2} : & t_{1} = 1 ∀t_{2} ∈ [0, 1], \\ l_{3} : & t_{2} = 1 ∀t_{1} ∈ [−2, 1]), \\ l_{4} : & t_{2} = 0 ∀t_{1} ∈ [−1 − i, 1]. \end{matrix}

Let C = {(t_{1}, 0) : t_{1} ∈ [−2, 1]}. Clearly, x̄ = (0, 0) is a strong C-relative interior point of the system {C, C_{i} : i ∈ I}. Note that i_{0} = 0 is the only limit point of I and that C_{0} is the rectangle [−2, 1]×[0, 1]. Hence it is easy to see that the set-valued function i ↦ C_{i} is lower semicontinuous on I. However, since (aff C) ∩ C_{i} is the segment [−1 − i, 1] × {0} for each i ∈ I \ {0} and (aff C) ∩ C_{0} = [−2, 1] × {0}, the set-valued function i ↦ (aff C) ∩ C_{i} is not lower semicontinuous on I. Let x_{0} = (−1, 0). It is clear that {aff C, C_{i}} has the strong CHIP at x_{0} for each i ∈ I. But the system {C, C_{i} : i ∈ I} does not have the strong CHIP at x_{0} because C ∩ (∩_{i∈I}C_{i}) = {(t_{1}, 0) : t_{1} ∈ [−1, 1]}, N_{C∩(∩_{i∈I}C_{i})}(x_{0}) = {(t_{1}, t_{2}) : t_{1} ≤ 0} and N_{C}(x_{0}) + ∑_{i∈I} N_{C_{i}}(x_{0}) = {(t_{1}, t_{2}) : t_{1} = 0}.

Remark 4.4. In general, {aff C, C_{1}} may not have the strong CHIP even if the C-relative interiorpoint condition is satisfied by the system {C, C_{1}}. For example, let I = {1}, X = ℝ^{2} and let C be the line t_{2} = 0 while C_{1} is bounded by the lines t_{2} = 0, t_{2} = 2, t_{1} = 1 and the half-circle t_{1}^{2} +(t_{2} −1)^{2} = 1, t_{1} ∈ [−1, 0]. Clearly, {C, C_{1}} satisfies the the C-relative interior-point condition but {aff C, C_{1}} does not have the strong CHIP at x_{0} = (0, 0). This example also shows that (c) cannot be dropped in the first part of Theorem 4.1.

Our next pair of theorems address the case when some C_{i} might have an empty interior. We will use the notion that a subset in X is finite codimensional.

Definition 4.1. Let A and B be two nonempty convex subsets of X. We say that A is

(i) finite codimensional in B if the closed subspace \overline{span B} ∩ \overline{span A} is a finite codimensional subspace of \overline{span B};

(ii) finite codimensional if \overline{span A} is finite codimensional in X.

Obviously, if B is finite dimensional, any nonempty convex subset A of X is finite codimensional in B.

Theorem 4.2. Let x_{0} ∈ C ∩ ( ∩_{i∈I} C_{i}). The system {C, C_{i} : i ∈ I} has the strong CHIP at x_{0} if the following conditions are satisfied.

(a) The system {C, C_{i} : i ∈ I} satisfies the weak-strong C-relative interior-point condition with (I_{1}, I_{2}).

(b) The set-valued mapping i ↦ (aff C) ∩ C_{i} is lower semicontinuous on I.

(c) The pair {aff C, C_{i}} has the strong CHIP at x_{0} for each i ∈ I \ (I_{1} ∪ I_{2}).

(d) C is finite dimensional or I_{C}^{rb}(x_{0}) is finite.

(e) C_{i} is finite codimensional for each i ∈ I_{1} Moreover, the same conclusion also holds if (a), (b) and (c) in the above assumptions are replaced

simultaneously by (a^{∗}) and (b^{∗}), where (a^{∗}) Same as (a) but delete the word “relative”. (b^{∗}) The set-valued mapping i ↦ C_{i} is lower semicontinuous on I.

Proof. By (a), there exist x̄ ∈ X and two disjoint finite subsets I_{1}, I_{2} of I such that x̄ is a weak-strong C-relative interior point with (I_{1}, I_{2}) of the CCS-system {C, C_{i} : i ∈ I}, that is, C_{i} is a polyhedron for each i ∈ I_{2} and

x̄ ∈ ri C ⋂ (rint_{C} ⋂_{i∈I_{0}} C_{i}) ⋂ ( ⋂_{i∈I_{1}} ri C_{i}) ⋂ ( ⋂_{i∈I_{2}} C_{i}) ,  (4.11)

where I_{0} = I \ (I_{1} ∪ I_{2}). Let J denote the closure of I_{0} , namely J equals the union of I_{0} with I^{l}, where the subset I^{l} of I_{1} ∪ I_{2} is defined by

I^{l} := {i ∈ I_{1} ∪ I_{2} : i is a limit point of I}.  (4.12)

For each i ∈ I^{l}, we define C⃗_{i} by

C⃗_{i} : = {x ∈ X : ∃x_{j} ∈ (aff C) ∩ C_{j} ∀j ∈ I \ {i} such that lim_{j→i} x_{j} = x}
= {x ∈ X : ∃x_{j} ∈ (aff C) ∩ C_{j} ∀j ∈ I_{0} such that lim_{j→i} x_{j} = x}, (4.13)

thanks to the fact that I_{1} ∪ I_{2} is a finite set. By assumption (b) and Proposition 3.1, we have

(aff C) ∩ C_{i} ⊆ C⃗_{i} for each i ∈ I^{l}.  (4.14)

Moreover,

x̄ ∈ ri C ⋂ (rint_{C} ⋂_{i∈I_{0}} C_{i}) ⋂ (rint_{C} ⋂_{i∈I^{l}} C⃗_{i}) .  (4.15)

In fact, since x̄ ∈ rint_{C} ⋂_{i∈I_{0}} C_{i}, there exists δ > 0 such that (aff C) ∩ B(x̄, δ) ⊆ C_{j} for each j ∈ I_{0}. From (4.13) it follows that (aff C) ∩ B(x̄, δ) ⊆ C⃗_{i} for each i ∈ I^{l}. Therefore x̄ ∈ rint_{C} (⋂_{i∈I^{l}} C⃗_{i}) and (4.15) holds. Note further that each C⃗_{i} is convex and closed. In fact, let i ∈ I^{l} and let {x_{n}} ⊆ C⃗_{i} be such that x_{n} → z. Then lim_{j→i} d_{C_{j}∩aff C}(x_{n}) = 0 for each n = 1, 2, . . .. Since

d_{C_{j}∩aff C}(z) ≤ ∥x_{n} − z∥ + d_{C_{j}∩aff C}(x_{n}),

we have that lim_{j→i} d_{C_{j}∩aff C}(z) = 0. This implies that z ∈ C⃗_{i} and so C⃗_{i} is closed. The convexity of C⃗_{i} follows from the convexity of the sets (aff C) ∩ C_{j} (j ∈ I), and the definition of C⃗_{i}. Recall that J = I_{0} ∪I^{l} and define C⃗_{i} = C_{i} for each i ∈ I_{0}. Then J is compact and {C, C⃗_{i} : i ∈ J} is a CCS-system with the following properties:

(i) {C, C⃗_{i} : i ∈ J} satisfies the strong C-relative interior-point condition; in fact,

x̄ ∈ ri C ⋂ rint_{C} ( ⋂_{j∈J} C⃗_{j})  (4.16)

(seeing (4.15)).

(ii) the set-valued function j ↦ (aff C) ∩ C⃗_{j} is lower semicontinuous on J.

(iii) Either C is finite dimensional or I⃗_{C}^{rb}(x_{0}) is a finite set, where

I⃗_{C}^{rb}(x_{0}) := {i ∈ J : x_{0} ∈ bd_{C} C⃗_{i}}.  (4.17)

In fact, (ii) follows from assumption (b) and the definition of C⃗_{j}. Moreover, (iii) follows from (d) and the fact that I⃗_{C}^{rb}(x_{0}) ⊆ I_{C}^{rb}(x_{0}) (because of (4.14) and x_{0} ∈ (aff C) ∩ C_{i}). Thus, by Remark 4.2, we have that

N_{C∩(∩_{j∈J} C⃗_{j})}(x_{0}) ⊆ N_{C}(x_{0}) + ∑_{j∈J} N_{(aff C)∩C⃗_{j}}(x_{0}).  (4.18)

We will show that

N_{(aff C)∩C⃗_{j}}(x_{0}) ⊆ N_{C}(x_{0}) + N_{C_{j}}(x_{0}) for each j ∈ J.  (4.19)

This inclusion is simply from the assumption (c) if j ∈ I_{0}. Next consider the case when j ∈ J ∩ I_{2}. In this case one has j ∈ I^{l} ∩ I_{2} and it follows from (4.14) that (aff C) ∩ C_{j} ⊆ C⃗_{j}. Consequently, one has

N_{(aff C)∩C⃗_{j}}(x_{0}) ⊆ N_{(aff C)∩C_{j}}(x_{0}) ⊆ N_{C∩C_{j}}(x_{0}) = N_{C}(x_{0}) + N_{C_{j}}(x_{0}),

where the last equality holds by Theorem 2.2 (which is applicable here because C_{j} is a polyhedron and x̄ ∈ ri C ∩ C_{j}). It remains to consider the case when j ∈ I_{1} ∩ I^{l} for (4.19). To do this and for a latter application, let us consider a general j ∈ I_{1} (for the time being regardless whether j ∈ I^{l} or not). Then by (4.11) and definitions, x̄ ∈ ri C_{j} = rint_{C_{j}} C_{j}. Hence, by Lemma 3.1, C_{j} − x̄ admits an (C_{j})-Minkowski extension C̃_{j} − x̄: C̃_{j} is a closed convex set such that

x̄ ∈ int C̃_{j} and \overline{aff C_{j}} ∩ C̃_{j} = \overline{aff C_{j}} ∩ C_{j} = C_{j} = (aff C_{j}) ∩ C̃_{j} ∀j ∈ I_{1}.  (4.20)

Combining this with (4.14),

(aff C) ∩ \overline{aff C_{j}} ∩ C̃_{j} ⊆ (aff C) ∩ C⃗_{j} for each j ∈ I_{1} ∩ I^{l}.  (4.21)

Thus if j ∈ I_{1} ∩ I^{l}, then it follows from (4.21) and Theorem 2.2 that

N_{(aff C)∩C⃗_{j}}(x_{0}) ⊆ N_{aff C∩\overline{aff C_{j}}∩C̃_{j}}(x_{0})
⊆ N_{(aff C)∩\overline{aff C_{j}}}(x_{0}) + NC̃_{j}(x_{0})
⊆ N_{aff C}(x_{0}) + N\overline{aff C_{j}}(x_{0}) + NC̃_{j}(x_{0})
⊆ N_{aff C}(x_{0}) + N_{\overline{aff C_{j}}∩C̃_{j}}(x_{0})
⊆ N_{C}(x_{0}) + N_{C_{j}}(x_{0})

thanks to (4.20) and so (4.19) is verified. Here we have used Theorem 2.2 (twice) which is applicable as x̄ ∈ int C̃_{j} ∩\overline{aff C_{j}} ∩ri (aff C) and \overline{aff C_{j}} is a polyhedron (being an affine subspace of finite codimension) by assumption (e). Therefore (4.19) is established in all possible cases. Combining (4.19) and (4.18), we have

N_{C∩(∩_{j∈J} C⃗_{i})}(x_{0}) ⊆ N_{C}(x_{0}) + ∑_{j∈J} N_{C_{j}}(x_{0}).  (4.22)

Recalling C⃗_{i} = C_{i} for each i ∈ I_{0}, J = I_{0} ∪ I^{l} and I = J ∪ I_{1} ∪ I_{2}, it is a routine matter to verify from (4.14) and (4.20) that

C ⋂ (⋂_{i∈I} C_{i}) = \{C ⋂ (⋂_{i∈J} C⃗_{i}) ⋂ ( ⋂_{i∈I_{1}} \overline{aff C_{i}}) ⋂ ( ⋂_{i∈I_{2}} C_{i})\} ⋂ ( ⋂_{i∈I_{1}} C̃_{i}) .  (4.23)

(For example if x is a member of the set on the right-hand side of (4.23) and if i ∈ I_{1} then x ∈ C_{i} by (4.20) and so it is not difficult to verify that x belongs to the set on the left-hand side of (4.23).) Moreover by virtue of general inclusion property, ri C ⋂ (rint_{C} ⋂_{i∈J} C⃗_{i}) ⊆ ri (C ⋂ ⋂_{i∈J} C⃗_{i}) (which

can be verified by definition). This with (4.16) implies that x̄ ∈ ri (C ⋂ (⋂_{i∈J} C⃗_{i})) and hence it follows from (4.11) and (4.20) that

x̄ ∈ ri (C ⋂ (⋂_{i∈J} C⃗_{i})) ⋂ ( ⋂_{i∈I_{1}} \overline{aff C_{i}}) ⋂ ( ⋂_{i∈I_{2}} C_{i}) ⋂ (int ⋂_{i∈I_{1}} C̃_{i}) .  (4.24)

Thus Theorem 2.2 is applicable to compute the normal cone of the set on the left-hand side of (4.23) at x_{0} (noting that each C_{i} with i ∈ I_{2} is a polyhedron and that each \overline{aff C_{i}} with i ∈ I_{1} is also a polyhedron as noted before):

N(x_{0}; C ∩ (∩_{i∈I}C_{i})) = N(x_{0}; (C ∩ (∩_{i∈J} C⃗_{i})) ∩ (∩_{i∈I_{1}} \overline{aff C_{i}}) ∩ (∩_{i∈I_{2}}C_{i})) + ∑_{i∈I_{1}} N(x_{0}; C̃_{i})

= N(x_{0}; C ∩ (∩_{i∈J} C⃗_{i})) + ∑_{i∈I_{1}} N(x_{0}; \overline{aff C_{i}}) + ∑_{i∈I_{2}} N(x_{0}; C_{i}) + ∑_{i∈I_{1}} N(x_{0}; C̃_{i})

⊆ N(x_{0}; C) + ∑_{i∈J} N(x_{0}; C_{i}) + ∑_{i∈I_{1}} N(x_{0}; \overline{aff C_{i}} ∩ C̃_{i}) + ∑_{i∈I_{2}} N(x_{0}; C_{i}),

thanks to (4.22). Since I = J ∪ I_{1} ∪ I_{2} and in view of (4.20), this implies that {C, C_{i} : i ∈ I} has the strong CHIP at x_{0}. This completes the proof for the first part of Theorem 4.2.

For the proof of the second part, by (a^{∗}), there exist x̄ ∈ X and two disjoint finite subsets I_{1}, I_{2} of I such that

x̄ ∈ ri C ⋂ (int ⋂_{i∈I_{0}} C_{i}) ⋂ ( ⋂_{i∈I_{1}} ri C_{i}) ⋂ ( ⋂_{i∈I_{2}} C_{i})  (4.25)

Now the proof is completed almost the same as for the first part, with the only modifications as follows. We use (4.25) in place of (4.11). We use “int” to replace “rint_{C}” in (4.15); in (4.13) and (4.14), we use “C_{i}” to replace “(aff C) ∩ C_{i}”.

Below we provide a sufficient condition ensuring the strong CHIP for a CCS-system with a closed subspace as a base-set.

Lemma 4.1. Let {C_{i} : i ∈ I} be a family of nonempty closed convex subsets of X, Y ⊆ X a vector subspace and let x_{0} ∈ Y ⋂ (⋂_{i∈I} C_{i}). Then the system {Y, C_{i} : i ∈ I} has the strong CHIP at x_{0} provided that

(a) for each i ∈ I, either C_{i} ⊆ Y or C_{i} is a polyhedron;

(b) {Y ∩ C_{i} : i ∈ I} has the strong CHIP at x_{0} in Y.

Proof. By (b) and recalling the notation of N^{Y} given in (2.17), we have

N_{Y ∩(∩_{i∈I}C_{i})}^{Y}(x_{0}) = ∑_{i∈I} N_{Y ∩C_{i}}^{Y}(x_{0}).

Now let x^{∗} ∈ N_{Y ∩(∩_{i∈I}C_{i})}(x_{0}). Then x^{∗}|_{Y} ∈ N_{Y ∩(∩_{i∈I}C_{i})}^{Y}(x_{0}), and hence, there exist a finite subset I_{0} of I and x_{i}^{∗} ∈ N_{Y ∩C_{i}}^{Y}(x_{0}), i ∈ I_{0}, such that

x^{∗}|_{Y} = ∑_{i∈I_{0}} x_{i}^{∗} on Y.  (4.26)

For each i ∈ I_{0}, let x_{i}^{∗} ∈ X^{∗} be an extension of x_{i}^{∗}. Then x_{i}^{∗} ∈ N_{Y ∩C_{i}}(x_{0}). Note that, in the case when C_{i} ⊆ Y, N_{Y ∩C_{i}}(x_{0}) = N_{C_{i}}(x_{0}) while, in the case when C_{i} is a polyhedron, N_{Y ∩C_{i}}(x_{0}) = N_{Y} (x_{0})+N_{C_{i}}(x_{0}) by Theorem 2.2. Thus (a) implies that ∑_{i∈I_{0}} x_{i}^{∗} ∈ N_{Y} (x_{0})+∑_{i∈I} N_{C_{i}}(x_{0}). Denote y^{∗} := x^{∗} − ∑_{i∈I_{0}} x_{i}^{∗}. Then y^{∗} ∈ N_{Y} (x_{0}) and so x^{∗} = y^{∗} + ∑_{i∈I_{0}} x_{i}^{∗} ∈ N_{Y} (x_{0}) + ∑_{i∈I} N_{C_{i}}(x_{0}). This shows that N_{Y ∩(∩_{i∈I}C_{i})}(x_{0}) ⊆ N_{Y} (x_{0}) + ∑_{i∈I} N_{C_{i}}(x_{0}) and the proof is complete.

The following consequence of Lemma 4.1 and Theorem 4.2 will be further extended in Corollary 4.4.

Corollary 4.1. Let C, C_{1} be a pair of closed convex subsets of X such that ri C ∩ ri C_{1} ≠ ∅. Suppose that (at least) one of the sets is finite dimensional or finite codimensional. Then {C, C_{1}} has the strong CHIP.

Proof. By symmetry, we need only to consider two cases: (i) C_{1} is finite codimensional; (ii) C is finite dimensional. The case (i) is clear by Theorem 4.2 (with I = I_{1} = {1} and I_{2} = ∅). For the case (ii), let x_{0} ∈ C ∩ C_{1}. Let \overline{span C} + \overline{span C_{1}} be denoted by Y. Then C_{1} is finite codimensional in Y and hence, by what we just proved, the system {C, C_{1}} in Y has the strong CHIP in Y. Since C and C_{1} are subsets of Y, it follows from Lemma 4.1 that the system {Y, C, C_{1}} has the strong CHIP. This implies that {C, C_{1}} has the strong CHIP in X since N_{Y} (x_{0}) ⊆ N_{C}(x_{0}) for each x_{0} ∈ C ∩ C_{1}.

If the assumption (d) in Theorem 4.2 is replaced by the stronger assumption that C is finite dimensional then (e) can be dropped This will be proved in Theorem 4.3 below. For preparing its proof and also for a later use, we first give a lemma.

Lemma 4.2. Let x_{0} ∈ C ∩ ( ∩_{i∈I} C_{i}). The CCS-system {C, C_{i} : i ∈ I} has the strong CHIP at x_{0} if it satisfies (a^{∗}), (b^{∗}) of Theorem 4.2 as well as the following conditions.

(c̄) {C, C_{i} : i ∈ I_{1} ∪ I_{2}} has the strong CHIP at x_{0}.

(d) Same as (d) in Theorem 4.2.

Proof. As in the beginning of the proof of Theorem 4.2, let I^{l} be defined by (4.12) and let C⃗_{i}, for each i ∈ I^{l}, be defined by (4.13) with X in place of C. Let J := I_{0} ∪ I^{l}. Define C⃗_{i} = C_{i} for each i ∈ I_{0}. Then J is compact and {C, C⃗_{i} : i ∈ J} is a CCS-system with the properties:

(i) {C, C⃗_{i} : i ∈ J} satisfies the strong interior-point condition; in fact,

x̄ ∈ ri C ⋂ (int ⋂_{j∈J} C⃗_{j}) .  (4.27)

(ii) the set-valued function j ↦ C⃗_{j} is lower semicontinuous on J.

(iii) C_{j} ⊆ C⃗_{j} for each j ∈ J.

(iv) C is finite dimensional or J⃗(x_{0}) is finite, where

J⃗(x_{0}) = {i ∈ J : x_{0} ∈ bdC⃗_{i}}.

Thanks to (i), (ii) and (iv), It is easy to see that the system {C, C⃗_{i} : i ∈ J} satisfies the conditions (a^{∗}), (b^{∗}) and (d) of Theorem 4.1. Hence, by Theorems 2.2, 4.1 and the above (iii), we obtain that

\begin{aligned}N(x_{0}; C) + N(x_{0}; (∩_{i∈J} C⃗_{i})) & = N(x_{0}; C ∩ (∩_{i∈J} C⃗_{i})) \\ & = N(x_{0}; C) + ∑_{i∈J} N(x_{0}; C⃗_{i}) \\ & ⊆ N(x_{0}; C) + ∑_{i∈J} N(x_{0}; C_{i}). \end{aligned}  (4.28)

Noting by (iii) that

C ⋂ (⋂_{i∈I} C_{i}) = C ⋂ (⋂_{i∈J} C⃗_{i}) ⋂ ( ⋂_{i∈I_{1}∪I_{2}} C_{i}) ,  (4.29)

and that

x̄ ∈ ri C ⋂ (int ⋂_{j∈J} C⃗_{j}) ⋂ ( ⋂_{i∈I_{1}∪I_{2}} C_{i}) ,  (4.30)

(a) of Theorem 2.2 can be applied to conclude that

\begin{aligned}N(x_{0}; C ∩ (∩_{i∈I}C_{i})) & = N(x_{0}; ∩_{i∈J} C⃗_{i}) + N(x_{0}; C ∩ (∩_{i∈I_{1}∪I_{2}}C_{i}) \\ & = N(x_{0}; C) + N(x_{0}; (∩_{i∈J} C⃗_{i})) + ∑_{i∈I_{1}} N(x_{0}; C_{i}) + ∑_{i∈I_{2}} N(x_{0}; C_{i}), \end{aligned}

(4.31)

thanks to the assumption (c̄). Combining (4.28) and (4.31) gives the desired conclusion and the proof is complete.

Theorem 4.3. Let x_{0} ∈ C ∩ ( ∩_{i∈I} C_{i}). The system {C, C_{i} : i ∈ I} has the strong CHIP at x_{0} if it satisfies (a), (b), (c) of Theorem 4.2 and the following condition.

(d̄) C is finite dimensional.

Moreover, the same conclusion also holds if (a) + (b) + (c) + (d̄) in the above assumptions is replaced by (a^{∗}) + (b^{∗}) + (d̄), where (a^{∗}) and (b^{∗}) are as in Theorem 4.2.

Proof. Denote

Ĉ = C − x_{0}, Z := span Ĉ, Ĉ_{i} = C_{i} − x_{0} for each i ∈ I.  (4.32)

Then, by assumptions, Z is finite dimensional and

ri Ĉ ⋂ (rintĈ ⋂_{i∈I_{0}} Ĉ_{i}) ⋂ ( ⋂_{i∈I_{1}} ri Ĉ_{i}) ⋂ ( ⋂_{i∈I_{2}} Ĉ_{i}) ≠ ∅.  (4.33)

Letting C_{i}^{♯} denote the intersection Z ∩ Ĉ_{i}, it follows that

ri Ĉ ⋂ (rintĈ ⋂_{i∈I_{0}} C_{i}^{♯}) ⋂ ( ⋂_{i∈I_{1}} ri C_{i}^{♯}) ⋂ ( ⋂_{i∈I_{2}} C_{i}^{♯}) ≠ ∅.  (4.34)

Noting aff Ĉ = Z, this implies that, as a system in the Banach space Z, {Ĉ, C_{i}^{♯} : i ∈ I} satisfies the weak-strong interior-point condition. In fact, with respect to the relative topology in Z, one has

rint_{Z} Ĉ ⋂ (rint_{Z} ⋂_{i∈I_{0}} C_{i}^{♯}) ⋂ ( ⋂_{i∈I_{1}} ri C_{i}^{♯}) ⋂ ( ⋂_{i∈I_{2}} C_{i}^{♯}) ≠ ∅.  (4.35)

By Theorem DLW (applied to the system {Ĉ, ⋂_{i∈I} C_{i}^{♯}} in Z), we have

N_{Ĉ∩(∩_{i∈I}C_{i}^{♯})}^{Z}(0) = N Z ; Ĉ(0) + N_{∩_{i∈I}C_{i}^{♯}}^{Z}(0),

namely, in the normed linear space X,

N_{Ĉ∩(∩_{i∈I} Ĉ_{i})}(0) = NĈ(0) + N_{∩_{i∈I}C_{i}^{♯}}(0),  (4.36)

because Ĉ ∩ (∩_{i∈I}C_{i}^{♯}) = Ĉ ∩ (∩_{i∈I} Ĉ_{i}), and Ĉ as well as C_{i}^{♯} are subsets of Z. We will show that

N_{∩_{i∈I}C_{i}^{♯}}(0) ⊆ NĈ(0) + ∑_{i∈I} NĈ_{i}(0).  (4.37)

Granting this, (4.36) and an easy translation argument imply that

N_{C∩(∩_{i∈I}C_{i})}(x_{0}) ⊆ N_{C}(x_{0}) + ∑_{i∈I} N_{C_{i}}(x_{0}),

which shows that {C, C_{i} : i ∈ I} has the strong CHIP at x_{0}. It remains to prove (4.37). To do this, we shall apply Lemma 4.1 to Y := Z and the system {D_{i}} where

D_{i} = \{ \begin{matrix}C_{i}^{♯} = Ĉ_{i} ∩ Z, & i ∈ I \ I_{2}, \\ Ĉ_{i}, & i ∈ I_{2}. \end{matrix}

We suppose, without loss of generality, that I \ I_{2} ≠ ∅ (otherwise (4.37) holds by Theorem 2.2). Then ∩_{i∈I}C_{i}^{♯} = ∩_{i∈I}D_{i}. Moreover, the assumption (c) (for i ∈ I \ (I_{1} ∪ I_{2})) and Corollary 4.1 (for i ∈ I_{1} ) imply that, for each i ∈ I \ I_{2}, {Z, Ĉ_{i}} has the strong CHIP at 0 and hence that

N_{D_{i}}(0) = N_{Z∩Ĉ_{i}}(0) = N_{Z}(0) + NĈ_{i}(0) ⊆ NĈ(0) + NĈ_{i}(0).  (4.38)

We claim that {D_{i} : i ∈ I} has the strong CHIP at 0. Granting this, it follows from (4.38) that

N_{∩_{i∈I}C_{i}^{♯}}(0) = N_{∩_{i∈I}D_{i}}(0) = ∑_{i∈I} N_{D_{i}}(0) = ∑_{i∈I\I_{2}} N_{D_{i}}(0) + ∑_{i∈I_{2}} NĈ_{i}(0) ⊆ NĈ + ∑_{i∈I} NĈ_{i}(0)

which shows (4.37). Thus it remains to prove the above claim. In view of Lemma 4.1, it suffices to show that, as a system in the subspace Z, {Z ∩ D_{i} : i ∈ I} has the strong CHIP at 0 (note that, for each i ∈ I_{2}, Ĉ_{i} is a polyhedron because C_{i} is a polyhedron). By (4.35), this system in Z satisfies the weak-strong interior-point condition on Z with (I_{1}, I_{2}). The assumption (b) tells us that i ↦ D_{i} ∩ Z is lower semicontinuous on I. It is trivial that, for each i ∈ I_{1}, D_{i} ∩ Z is finite codimensional in Z as Z is finite dimensional. Hence it is easy to see that the second part of Theorem 4.2 is applicable to the system {Z, Z ∩ D_{i} : i ∈ I} in place of {C, C_{i} : i ∈ I} if the assumptions (a), (b), (c) and (d̄) are assumed. Thus, this system (in Z) does have the strong CHIP at 0. Our claim is therefore established and this completes the proof of the first part of the theorem.

For the second part, suppose that the system {C, C_{i} : i ∈ I} satisfies (a^{∗}) + (b^{∗}) + (d̄) (in place of (a) + (b) + (c) + (d̄)). Then, by (d̄) and by the result of the first part (applied to the finite subsystem {C, C_{i} : i ∈ I_{1} ∪ I_{2}}), one concludes that {C, C_{i} : i ∈ I_{1} ∪ I_{2}} has the strong CHIP at x_{0}. Hence, by Lemma 4.2, {C, C_{i} : i ∈ I} also has the strong CHIP at x_{0}.

Corollary 4.2. Let x_{0} ∈ C ∩ ( ∩_{i∈I} C_{i}). The system {C, C_{i} : i ∈ I} has the strong CHIP at x_{0} if it satisfies (a^{∗}), (b^{∗}) of Theorem 4.2 as well as the following condition.

(d^{∗}) At least one of {C, C_{i} : i ∈ I_{1}} is finite dimensional.

Proof. By assumption (a^{∗}), take x̄ satisfying (4.25). By assumption (d^{∗}) and in view of the second part of Theorem 4.3, it suffices to consider the case when C_{i_{0}} is finite dimensional for some i_{0} ∈ I_{1}. Let

I′_{1} = I_{1} ∪ {i_{∞}}

where i_{∞} is a new index such that i_{∞} ∉ I. Let I_{0} = I \ (I_{1} ∪ I_{2}) and define D := C_{i_{0}} ∩ (∩_{i∈I_{0}})C_{i}. Then ri C_{i_{0}} ∩ int (∩_{i∈I_{0}}C_{i}) ⊆ ri D an so x̄ ∈ ri D by (4.25). Thus

x̄ ∈ ri D ⋂ ( ⋂_{i∈I_{1}} ri C_{i}) ⋂ ri C ⋂ ( ⋂_{i∈I_{2}} C_{i}) .  (4.39)

Letting J = I′_{1} ∪ I_{2}, D_{i_{∞}} = C, and D_{i} = C_{i} for each i ∈ I_{1} ∪ I_{2}, (4.39) implies that the new CCSsystem {D, D_{j} : j ∈ J} satisfies the weak-strong interior-point condition with (I′_{1}, I_{2}), that is, the condition (a^{∗}) of Theorem 4.3 stated for {C, C_{i} : i ∈ I}. It also satisfies (b^{∗}) of Theorem 4.3 as J is finite. Therefore, applying Theorem 4.3 to the new system, we have that

\begin{aligned}N_{D∩(∩_{j∈J} D_{i})}(x_{0}) & = & N_{D}(x_{0}) + ∑_{j∈J} N_{D_{j}}(x_{0}) \\ & = & N_{C_{i_{0}} ∩(∩_{i∈I_{0}} C_{i})}(x_{0}) + N_{C}(x_{0}) + ∑_{i∈I_{1}∪I_{2}} N_{C_{i}}(x_{0}) \\ & ⊆ & N_{C}(x_{0}) + N_{C_{i_{0}} ∩(∩_{i∈I}C_{i})}(x_{0}). \end{aligned}  (4.40)

Applying Theorem 4.3 to the system {C_{i_{0}}, C_{i} : i ∈ I} and noting that D ∩ (∩_{j∈J}D_{i}) = C ∩ (∩_{i∈I}C_{i}), it follows from (4.40) that

N_{C∩(∩_{i∈I}C_{i})}(x_{0}) = N_{D∩(∩_{j∈J} D_{i})}(x_{0}) = N_{C}(x_{0}) + ∑_{i∈I} N_{C_{i}}(x_{0}).

The proof is complete.

Corollary 4.3. Let x_{0} ∈ C ∩ ( ∩_{i∈I} C_{i}). The system {C, C_{i} : i ∈ I} has the strong CHIP at x_{0} if it satisfies (a^{∗}), (b^{∗}) of Theorem 4.2 as well as the following conditions.

(c^{∗}) For any i, j ∈ I_{1}, C_{i} is finite codimensional in C as well as in C_{j}.

(d̃) I(x_{0}) = {i ∈ I : x_{0} ∈ bd C_{i}} is finite.

Proof. By Lemma 4.2, it suffices to show that (c̄) of Lemma 4.2 holds, that is, the system {C, C_{i} : i ∈ I_{1} ∪ I_{2}} has the strong CHIP at x_{0}. For this purpose, take a new index 0 ∉ I and, for each i ∈ I_{1}, let Y_{0} and Y_{i} denote the closed subspaces spanned by C and C_{i}, respectively, and let

Y = Y_{0} + ∑_{i∈I_{1}} Y_{i}.  (4.41)

We claim that each C_{i} for i ∈ I_{1} is finite codimensional in Y. To verify this claim, let i ∈ I_{1} and set J = {0} ∪ I_{1} \ {i}. By assumption (c^{∗}), C_{i} is finite codimensional in Y_{j}, for each j ∈ J, and it follows from Definition 4.1 that there exists a finite dimensional subspace Y′_{j} of Y_{j} such that Y_{j} is the direct sum of Y′_{j} and Y_{j} ∩ Y_{i}, that is,

Y_{j} = Y′_{j} + (Y_{j} ∩ Y_{i}) and Y′_{j} ∩ (Y_{j} ∩ Y_{i}) = {0}.

Hence, for each j ∈ J,

Y_{j} + Y_{i} = Y′_{j} + Y_{i} and Y′_{j} ∩ Y_{i} = {0}.  (4.42)

Then, by (4.41) and (4.42),

Y = ∑_{j∈J}(Y_{j} + Y_{i}) = ∑_{j∈J} Y′_{j} + Y_{i} and (∑_{j∈J} Y′_{j}) ∩ Y_{i} = {0}.  (4.43)

This implies that Y_{i} is finite codimensional in Y since ∑_{j∈J} Y′_{j} is finite dimensional. The claim is proved. Noting that C_{i} ∩ Y = C_{i} for each i ∈ I_{1}, this implies that, as a CCS-system in Y, {C, C_{i} ∩ Y : i ∈ I_{1} ∪ I_{2}} has the property (e) of Theorem 4.2 stated for {C, C_{i} : i ∈ I}. Moreover it also satisfies (d) thanks to the assumption (d̃) and (4.1). Therefore one can apply the second part of Theorem 4.2 (with I_{0} = ∅) to conclude that this finite system in the subspace Y has the strong CHIP at x_{0}. Noting that C, C_{i} ⊆ Y for each i ∈ I_{1} and C_{j} is a polyhedron for each j ∈ I_{2}, it follows from Lemma 4.1 that {Y, C, C_{i} : i ∈ I_{1} ∪ I_{2}} has the strong CHIP at x_{0} in X. Therefore, {C, C_{i} : i ∈ I_{1} ∪ I_{2}} has the strong CHIP at x_{0} (because C ⊆ Y).

We obtain below an extension of Rockafellar’s result [28, Corollary 23.8.1, p.223] in the setting of general normed linear spaces.

Corollary 4.4. Let I = J ∪ K be finite with J, K disjoint such that C_{k} is a polyhedron for each k ∈ K, and suppose that

ri C ⋂ ( ⋂_{j∈J} ri C_{j}) ⋂ ( ⋂_{k∈K} C_{i}) ≠ ∅.  (4.44)

Then the system {C, C_{i} : i ∈ I} has the strong CHIP if at least one of the following conditions is satisfied.

(a) At least one of {C, C_{j} : j ∈ J} is finite dimensional.

(b) For each j ∈ J, C_{j} is finite codimensional respectively in C, and in C_{i} for each i ∈ J (e.g., C_{j} is finite codimensional for each j ∈ J).

Proof Since I is finite and thanks to (4.44), the system {C, C_{i} : i ∈ I} satisfies (a*) and (b*) of Theorem 4.2 (with (I_{1}, I_{2}) = (J, K)). Now apply Corollary 4.2 and Corollary 4.3 to conclude the proof.

5 Subsystems and the strong CHIP

Recall that I and {C, C_{i} : i ∈ I} are as explained in the beginning of the preceding section. Our main result of this section is the following.

Theorem 5.1. Suppose that the finite subsystem {C, C_{i} : i ∈ J} has the strong CHIP for each finite subset J of I. Then the system {C, C_{i} : i ∈ I} has the strong CHIP provided the following conditions are satisfied.

(a) C is finite dimensional (say, dim C := l < +∞).

(b) The set-valued function i ↦ C_{i} is Kuratowski continuous on I.

(c) For any finite subset J of I (with the number of elements |J| ≤ l if X is real, and |J| ≤ 2l if X is complex), the subsystem {C, C_{i} : i ∈ J} satisfies the C-relative interior-point condition:

C ⋂ (⋂_{i∈J} rint_{C}C_{i}) ≠ ∅.  (5.1)

Proof. Since I is compact and metrizable, there exists a sequence {I_{k}} of subsets of I such that (i) each I_{k} is finite; (ii) I_{k} ⊆ I_{k+1} for k = 1, 2, …;

(iii) I equals the closure of ⋃_{k=1}^{∞} I_{k}.

Let S = ⋂_{i∈I} C_{i}, K = C ∩ S and S_{k} = ∩_{i∈I_{k}}C_{i} for each k. Then

K = C ⋂ (⋂_{i∈I} C_{i}) ⊆ C ∩ S_{k}.  (5.2)

Let x_{0} ∈ C ⋂(⋂_{i∈I} C_{i}) and let x^{∗} ∈ N_{K}(x_{0}). We have to show that x^{∗} ∈ N_{C}(x_{0}) + ∑_{i∈I} N_{C_{i}}(x_{0}). We will first show that there exist {x_{k}} ⊆ C with x_{k} → x and x_{k}^{∗} ∈ N_{C∩S_{k}}(x_{k}) such that {x_{k}} is bounded and

lim_{k→∞}⟨x_{k}^{∗}, y⟩ = ⟨x^{∗}, y⟩ for each y ∈ span (C − x_{0}).  (5.3)

In fact, since Z := span (C − x_{0}) is finite dimensional, we may assume, without loss of generality, that the norm restricted to Z is both strictly convex and smooth. Clearly, we may assume that x^{∗}|_{Z} ≠ 0. Take z_{0} ∈ Z such that ⟨x^{∗}, z_{0}⟩ = ∥x^{∗}|_{Z}∥ · ∥z_{0}∥ = ∥z_{0}∥^{2}. Write x := x_{0} + z_{0}. Then, x^{∗}|_{Z} = J(x − x_{0})|_{Z}. By the Hahn-Banach Theorem, let x̄^{∗} ∈ X^{∗} be a norm-preserving extension of x^{∗}|_{Z}. Then x̄^{∗} ∈ N_{K}(x_{0}) ∩ J(x − x_{0}). Hence by Proposition 2.1, x_{0} = P_{K}(x). Let x_{k} = P_{C∩S_{k}}(x). Then x − x_{k} ∈ Z because x_{0} − x_{k}, x − x_{0} ∈ Z. Moreover,

∥x_{k}∥ ≤ ∥x_{k} − x∥ + ∥x∥ ≤ ∥x − x_{0}∥ + ∥x∥;

hence {x_{k}} ⊂ C is bounded. Without loss of generality, assume that x_{k} → x̄ for some x̄ ∈ C. Let i ∈ I. By (ii) and (iii), there exists {i_{k}} ⊆ I with i_{k} ∈ I_{k} for each k such that i_{k} → i. Noting that x_{k} ∈ C_{i_{k}} and that x_{k} → x̄, we have that x̄ ∈ C_{i} by the upper Kuratowski semicontinuity assumed in (b). This shows that x̄ ∈ K. Because

∥x − x̄∥ = lim_{k→∞} ∥x − x_{k}∥ ≤ ∥x − y∥ for each y ∈ K,

x̄ = P_{K}(x) = x_{0} and hence x_{k} → x_{0}. On the other hand, by Proposition 2.1, there exists x_{k}^{∗} ∈ N_{C∩S_{k}}(x_{k}) ∩ J(x − x_{k}). Consequently, {x_{k}^{∗}} is bounded since ∥x_{k}^{∗}∥ = ∥x − x_{k}∥. Moreover, by the smoothness of the norm in Z, the mapping z ↦ J(z)|_{Z} is norm-weak^{∗} continuous. Hence x_{k}^{∗}|_{Z} → x^{∗}|_{Z} as x − x_{k} → x − x_{0} in Z. Thus (5.3) holds.

By assumption, the finite subsystem {C, C_{i} : i ∈ I_{k}} has the strong CHIP at x_{k}, and hence

x_{k}^{∗} ∈ N_{C∩S_{k}}(x_{k}) = N_{C}(x_{k}) + ∑_{i∈I_{k}} N_{C_{i}}(x_{k}).  (5.4)

If there exists a subsequence {k_{j}} of {k} such that x_{k_{j}}^{∗} ∈ N_{C}(x_{k_{j}}) for each j, then, x^{∗} ∈ N_{C}(x_{0}) by (5.3) (and so x^{∗} ∈ N_{C}(x_{0}) + ∑_{i∈I} N_{C_{i}}(x_{0})). Therefore we may assume that, for each k, x_{k}^{∗} ∉ N_{C}(x_{k}). Recalling that the dimension of Z is l, it follows from (5.4) and [28, Corollary 17.1.2] that there exist z_{k}^{∗} ∈ N_{C}(x_{0}), i_{j}^{k} ∈ I_{k} and y_{i_{j}^{k}}^{∗} ∈ N_{C_{i^{k}_{j}}} (x_{0}) such that

x_{k}^{∗} = z_{k}^{∗} + ∑_{j=1}^{s} y_{i_{j}^{k}}^{∗} on Z for all k = 1, 2, …,  (5.5)

where s ≤ l if X is real, and s ≤ 2l if X is complex. Without loss of generality, assume that y_{i_{j}^{k}}^{∗}|_{Z} ≠ 0

for each j = 1, 2, …, s. Set λ_{j}^{k} = ∥y_{i_{j}^{k}}^{∗}|_{Z}∥, y_{i_{j}^{k}}^{∗} = ^{\frac{y_{i^{k}_{j}}^{∗}}{λ_{j}^{k}}}. Then λ_{j}^{k} > 0 for j = 1, 2, …, s and

x_{k}^{∗} = z_{k}^{∗} + ∑_{j=1}^{s} λ_{j}^{k}y_{i_{j}^{k}}^{∗} on Z for all k = 1, 2, ….  (5.6)

Let λ^{k} := ∑_{j=1}^{s} λ_{j}^{k}. Then {λ^{k}} is bounded. Indeed, if not, by considering a subsequence if necessary, we have that lim_{k→∞} λ^{k} = +∞. Thus \frac{x_{k}^{∗}}{λ^{k}} → 0 as k → ∞. Furthermore, without loss of generality, we may assume that, as k → ∞,

i_{j}^{k} → i_{j} and \frac{λ_{j}^{k}}{λ^{k}} → μ_{j}, j = 1, 2, …, s.  (5.7)

Then ∑_{j=1}^{s} μ_{j} = 1. Since \{y_{i_{j}^{k}}^{∗}\} is bounded, by (5.6), \{\frac{z_{k}^{∗}}{λ^{k}} |_{Z}\} is bounded too. Thus, we may also assume that there exist z_{0}^{∗}, y_{i_{j}}^{∗} ∈ Z^{∗} such that

\frac{z_{k}^{∗}}{λ^{k}} → z_{0}^{∗} and y_{i_{j}^{k}}^{∗} → y_{i_{j}}^{∗} on Z  (5.8)

as k → ∞. Then, since z_{k}^{∗} ∈ N_{C}(x_{k}), by (5.8) and the fact that x_{k} → x_{0},

⟨z_{0}^{∗}, z − x_{0}⟩ ≤ 0 for each z ∈ C.  (5.9)

Let z ∈ (Z + x_{0}) ∩ C_{i_{j}}. Since i_{j}^{k} → i_{j} and thanks to the assumption (b), there exists {z_{k}} with each z_{k} ∈ C_{i_{j}^{k}} such that z_{k} → z as k → ∞. Then ⟨y_{i_{j}^{k}}^{∗}, z_{k} − x_{k}⟩ ≤ 0. Since x_{k} → x_{0}, it follows from (5.8) that, for each j,

⟨y_{i_{j}}^{∗}, z − x_{0}⟩ ≤ 0 for each z ∈ (Z + x_{0}) ∩ C_{i_{j}}.  (5.10)

Consequently, by the Hahn-Banach Theorem, z_{0}^{∗} and y_{i_{j}}^{∗} can be extended to z_{0}^{∗} ∈ N_{C}(x_{0}) and y_{i_{j}}^{∗} ∈ N_{C_{i_{j}} ∩(Z+x_{0})}(x_{0}). Clearly, by (5.6), (5.7) and (5.8),

0 = z_{0}^{∗} + ∑_{j=1}^{s} μ_{j}y_{i_{j}}^{∗} on Z.  (5.11)

By the assumption (c), there exists ȳ ∈ C ∩ rint_{C}C_{i_{j}} for each j = 1, 2, …, s. Then, by (5.10), ⟨y_{i_{j}}^{∗}, ȳ − x_{0}⟩ ≤ 0 for each j = 1, 2, …, s. We claim that each of above inequalities must be strict. Indeed, suppose otherwise that ⟨y_{i_{j}}^{∗}, ȳ−x_{0}⟩ = 0 for some j. Let z ∈ Z and let z_{t} := t(z +x_{0})+(1−t)ȳ. Then for any t with |t| small enough, z_{t} ∈ C_{i_{j}} and so, by (5.10),

t⟨y_{i_{j}}^{∗}, z⟩ = ⟨y_{i_{j}}^{∗}, z_{t} − x_{0}⟩ − (1 − t)⟨y_{i_{j}}^{∗}, ȳ − x_{0}⟩ = ⟨y_{i_{j}}^{∗}, z_{t} − x_{0}⟩ ≤ 0.

This implies that ⟨y_{i_{j}}^{∗}, z⟩ = 0, that is, y_{i_{j}}^{∗}|_{Z} = 0, which contracts that ∥y_{i_{j}}^{∗}|_{Z}∥ = ∥y_{i_{j}}^{∗}∥ = 1. Hence,

⟨z_{0}^{∗} + ∑_{j=1}^{s} μ_{j}y_{i_{j}}^{∗}, ȳ − x_{0}⟩ < 0,

which contradicts (5.11). This shows that {λ^{k}} is bounded. Thus, taking the limits on the two sides of (5.6) and using the similar arguments as above (if necessary, use subsequences), we get that

x^{∗} = z_{0}^{∗} + ∑_{j=1}^{s} λ_{j}y_{i_{j}}^{∗} on Z  (5.12)

for some λ_{j} ≥ 0, z_{0}^{∗} ∈ N_{C}(x_{0}) and y_{i_{j}}^{∗} ∈ N_{C_{i_{j}}} (x_{0}) (j = 1, 2 …, s). Let y^{∗} = x^{∗} − z_{0}^{∗} − ∑_{j=1}^{s} λ_{j}y_{i_{j}}^{∗}. Then y^{∗} ∈ N_{C}(x_{0}) by (5.12) and so x^{∗} ∈ N_{C}(x_{0}) + ∑_{i∈I} N_{C_{i}}(x_{0}). The proof is complete.

Corollary 5.1. Suppose that the CCS-system {C, C_{i} : i ∈ I} satisfies the interior-point condition, dim C < +∞ and that the set-valued function i ↦ C_{i} is Kuratowski continuous. Then the system {C, C_{i} : i ∈ I} has the strong CHIP.

Proof. The assumed interior-point condition clearly implies (c) of Theorem 5.1; it also implies that each of the finite subsystems of {C, C_{i} : i ∈ I} has the strong CHIP by Theorem DLW. Hence the conclusion holds by Theorem 5.1.

Remark 5.1. Examples 5.1, 5.2 and 5.3 will show that none of the conditions (a), (b) and (c) in Theorem 5.1 can be dropped. Each of these examples will be a CCS-system without the strong CHIP, but each of the finite subsystems of each of these CCS-systems does have the strong CHIP (each C_{i} being a polyhedron, and the base-set being the whole space). In each of these examples, I is the compact subset of ℝ defined by I = \{0, 1, \frac{1}{2}, …, \frac{1}{i} …\}.

Example 5.1. Let C = X = {x = (x_{1}, x_{2}, …, x_{k}, …) : x_{k} ∈ ℝ, lim_{k} x_{k} exists} with the norm defined by

∥x∥ = sup_{k} |x_{k}|, x = (x_{k}) ∈ X.

Define

C_{i} = \{ \begin{matrix}\{x = (x_{k}) ∈ X : lim_{k} x_{k} ≤ 0\} & i = 0, \\ \{x = (x_{k}) ∈ X : x_{\frac{1}{i}} ≤ 0\} & i ∈ I \ \{0\}, \end{matrix} .

Then the set-valued function i ↦ C_{i} is Kuratowski continuous on I. In fact, let {i_{n}} ⊆ I be a sequences satisfying i_{n} → 0. To show the Kuratowski continuity at 0, let {x^{i_{n}}} be a sequences satisfying x^{i_{n}} ∈ C_{i_{n}} and x^{i_{n}} → x^{0}. Noting

x0_{\frac{1}{in}} = x0_{\frac{1}{in}} − xi_{n}_{\frac{1}{in}} + xi_{n}_{\frac{1}{in}} ≤ x0_{\frac{1}{in}} − xi_{n}_{\frac{1}{in}} ≤ ∥x^{0} − x^{i_{n}}∥ → 0,

One has that lim_{k} x_{k}^{0} ≤ 0 and so x^{0} ∈ C_{0}. This proves that the set-valued function i ↦ C_{i} is upper Kuratowski continuous at 0. To show the lower Kuratowski continuity at 0, let x^{0} ∈ C_{0}. Then a = lim_{k} x_{k}^{0} ≤ 0. Define x^{i_{n}} = (x_{k}^{i_{n}}) with x_{k}^{i_{n}} = x_{k}^{0} if k ≠ \frac{1}{i_{n}} and x_{k}^{i_{n}} = a if k = \frac{1}{i_{n}}. Then x^{i_{n}} ∈ C_{i_{n}} and lim_{n} x^{i_{n}} = x^{0}. This shows that the set-valued function i ↦ C_{i} is lower Kuratowski continuous at 0. Note that x̄ ∈ int(⋂_{i∈I} C_{i}) for x̄ = (−1, −1, …) ∈ X. Hence the conditions (b) and (c) in Theorem 5.1 are satisfied. Let x_{0} = 0. Then x_{0} ∈ ⋂_{i∈I} C_{i}. It is easy to see that ∑_{i∈I} N_{C_{i}}(x_{0}) is not closed; hence this system does not have the strong CHIP at x_{0}.

Example 5.2. Let C = X = ℝ^{2}. Define

C_{i} = \{ \begin{matrix}\{x = (x_{1}, x_{2}) ∈ X : x_{1} + x_{2} ≤ 0\} & i = 0, \\ \{x = (x_{1}, x_{2}) ∈ X : x_{1} + ix_{2} ≤ 0\} & i ∈ I \ \{0\}. \end{matrix}

Then ⋂_{i∈I} C_{i} = {(x_{1}, x_{2}) : x_{1} ≤ 0, x_{1} + x_{2} ≤ 0}. Let x_{0} = 0. Then x_{0} ∈ bd ⋂_{i∈I} C_{i}. Clearly,

(a) in Theorem 5.1 is satisfied. Since x̄ = (−1, \frac{1}{2})) ∈ int(⋂_{i∈I} C_{i}), the condition (c) in Theorem 5.1 is satisfied too. However, N_{⋂_{i∈I} C_{i}}(x_{0}) = {(t_{1}, t_{2}) : 0 ≤ t_{2} ≤ t_{1}} and ∑_{i∈I} N_{C_{i}}(x_{0}) = {(t_{1}, t_{2}) : 0 < t_{2} ≤ t_{1}} ∪ {(0, 0)}. Therefore this system does not have the strong CHIP at x_{0}. Note that the condition (b) is not satisfied.

Example 5.3. Let C = X = ℝ^{2} and define

C_{i} = \{ \begin{matrix}\{x = (x_{1}, x_{2}) : x_{2} ≤ 0\} & i = 1, \\ \{x = (x_{1}, x_{2}) : ix_{1} − x_{2} − i^{2} ≤ 0\}, & i ∈ I \ \{1\}. \end{matrix}

Then ⋂_{i∈I} C_{i} = {(x_{1}, 0) ∈ ℝ^{2} : x_{1} ≤ 0}. Let x_{0} = 0. Hence

N_{⋂_{i∈I} C_{i}}(x_{0}) = {(t_{1}, t_{2}) ∈ ℝ^{2} : t_{1} ≥ 0}

and

∑_{i∈I} N_{C_{i}}(x_{0}) = cone {(0, −1), (0, 1)} = {(t_{1}, t_{2}) ∈ ℝ^{2} : t_{1} = 0}}.

Consequently, this system does not have the strong CHIP at x_{0}. Note that the conditions (a) and (b) in Theorem 5.1 are satisfied but the condition (c) is not.

Acknowledgments. We thank the referee for his helpful comments.

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