On Best Restricted Range Approximation in Continuous Complex-valued Function 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. To provide a Kolmogorov type condition for characterizing a best approximation in a continuous complex-valued function space, it is usually assumed that the family of closed convex sets in the complex plane used to restrict the range satisfies a strong interior-point condition, and this excludes the interesting case when some Ω_{t} is a line-segment or a singleton. The main aim of the present paper is to remove this restriction by virtue of a study of the notion of the strong CHIP for infinite system of closed convex sets in a continuous complex-valued function space.

1 Introduction

Throughout this paper C(Q) denotes the Banach space of all complex-valued continuous functions on a compact metric space Q endowed with the uniform norm (the “Sup-norm”). Let P denote a finite dimensional subspace of C(Q), and let {Ω_{t} : t ∈ Q} be a family of nonempty closed convex sets in the complex plane ℂ. Set

P_{Ω} = {p ∈ P : p(t) ∈ Ω_{t} for each t ∈ Q}.  (1.1)

The captioned problem is that of finding an an element p^{∗} ∈ P_{Ω} for a function f ∈ C(Q) such that

∥f − p^{∗}∥ = inf_{p∈P_{Ω}} ∥f − p∥  (1.2)

∗The first author was supported in part by the National Natural Science Foundation of China (grant 10271025). The second author was supported by an Earmarked Grant from the Research Grant Council of Hong Kong.

(such p^{∗} is called a best restricted range approximation to f from P with respect to {Ω_{t}}). This problem was first presented and formulated by Smirnov and Smirnov in [24, 25]; their approach followed the standard path for the corresponding issue in the real-valued continuous function space theory. In [24], while it was pointed out that this problem for the general class of restrictions was quite difficult, they took up the special case when each Ω_{t} is a disk in ℂ. Later in a series of papers by themselves and by others [26, 27, 28, 11, 14], more general class of {Ω_{t}} has been considered but each of them is still under a general strong interior-point condition assumption that there exists an element p̄ of P_{Ω} such that int ∩_{t∈Q} (Ω_{t} − p̄(t)) ≠ ∅ (hence 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. Our results in Section 3 further relax the restriction by allowing the interesting case just mentioned. Letting

C_{t} = {u ∈ C(Q) : u(t) ∈ Ω_{t}} for each t ∈ Q,  (1.3)

we note that {P, C_{t} : t ∈ Q} is a family of closed convex sets in C(Q) with the intersection P ⋂(⋂_{t∈Q} C_{t}) = P_{Ω}. The main aim of this paper is to provide some characterizations for p^{∗} to satisfy (1.2) in a reasonable case (under appropriate continuity assumption of the set-valued mapping t ↦ Ω_{t}, and a suitably relaxed interior-point condition). One such a characterization is given, as in the corresponding real case, by a condition of the Kolmogorov type. Our results here are obtained by virtue of a study of the strong CHIP (the strong conical hull intersection property) for an infinite family of closed convex sets in a Banach space. The notion of the strong CHIP was first introduced by Deutsch, Li and Ward in [7, 8] for a finite family of closed convex sets in an Euclidean space (or a Hilbert space) and recently extended by Li and Ng in [14] to an arbitrary family of closed convex sets in a Banach space. In [16], this notion was studied extensively and some useful sufficient conditions for the strong CHIP were established.

We end this introduction with a short remark that having obtained the characterization results as presented in Section 3, the issue on the uniqueness of solutions for the corresponding problems can be addressed along a well established path (cf. [11]) and need not further elaborate here.

2 Notations and preliminary results

We begin with the notations used in this paper, most of which are standard (cf. [5, 10]). In particular, we assume that X is a complex (or real at times) Banach space. 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); the normal cone of Z at z_{0} is denoted by N_{Z}(z_{0}) and defined by

N_{Z}(z_{0}) = {x^{∗} ∈ X^{∗} : Re ⟨x^{∗}, z − z_{0}⟩ ≤ 0 for each z ∈ Z}.  (2.1)

Our main tools are the following Theorems 2.1 and 2.2 taken from [16, Corollary 4.2 and 5.1]. It would be convenient for us to repeat some of the definitions introduced in [16] as well as some other more standard notions in this regard. Let I denote an index-set which is assumed to be a compact metric space. 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 2.1. A CCS-system {C, C_{i} : i ∈ I} (with base-set C) is said to satisfy:

(i) the interior-point condition if

C ⋂ (⋂_{i∈I} int C_{i}) ≠ ∅;  (2.2)

(ii) the strong interior-point condition if

C ⋂ (int ⋂_{i∈I} C_{i}) ≠ ∅;  (2.3)

(iii) the weak-strong 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 ⋂ (int ⋂_{i∈I\(I_{1}∪I_{2})} C_{i}) ⋂ ( ⋂_{i∈I_{1}} ri C_{i}) ⋂_{i∈I_{2}} C_{i} ≠ ∅.  (2.4)

(iv) any point x̄ belonging to the set on the left-hand side of (2.2) (resp. (2.3), (2.4)) is called 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}.

It is trivial that (2.2)⟹(2.3). The converse also holds in some cases, one of which will be described in terms of continuity of some set-valued functions (cf. [16]). For set-valued functions there are many different notions of continuity. In Definitions 2.2 and 2.3 below, we recall two frequently used ones. We assume that Q is a compact metric space.

Definition 2.2. Let F : Q → 2^{X} be a set-valued function defined on Q and let t_{0} ∈ Q. Then F is said to be

(i) lower semicontinuous at t_{0}, 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) upper semicontinuous at t_{0} 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}).

(iii) lower (resp. upper) semicontinuous on Q if it is lower (resp. upper) semicontinuous at each t ∈ Q.

Definition 2.3. (cf.[23, p.55]) Let F : Q → 2^{X} 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 2.1. 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 ∪_{t∈Q}F (t) is compact.

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)

Definition 2.4. 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).  (2.6)

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

Theorem 2.1. 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 interior-point condition with (I_{1}, I_{2}).

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

(c) At least one of the sets in the family {C, C_{i} : i ∈ I_{1}} is finite-dimensional.

Theorem 2.2. 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.

We end this section with two results on characterizations of the strong CHIP of a (possibly infinite) system {C, C_{i} : i ∈ I} of closed convex sets. The first result, which is valid in general Banach space and will be used in next section, is given in terms of the optimality conditions of constrained best approximation while the second result in the Hilbert space setting is given as a dual formulation of constrained best approximation (see for example, [3, 4, 7, 8, 9, 12, 13, 14, 15, 17, 18]). For this end, we need a well-known result on the characterization of the best approximation by a convex set in X, which was established independently by Deutsch [6] and Rubenstein [20] (see also [1]). 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∥}.  (2.7)

Proposition 2.1. Let W be a closed 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}).

Theorem 2.3. Let K = C ∩ (∩_{i∈I}C_{i}) and x_{0} ∈ K. Consider the following statements.

(i) The system {C, C_{i} : i ∈ I} has the strong CHIP at x_{0}.

(ii) For each x ∈ X, x_{0} ∈ P_{K}(x) if and only if

J(x − x_{0}) ⋂ (N_{C}(x_{0}) + ∑_{i∈I} N_{C_{i}}(x_{0})) ≠ ∅.  (2.8)

(iii) For each x ∈ X, x_{0} ∈ P_{K}(x) if and only if

J(x − x_{0})|_{C−x_{0}} ⋂ (N_{C}(x_{0})|_{C−x_{0}} + ∑_{i∈I} N_{C_{i}}(x_{0})|_{C−x_{0}}) ≠ ∅.  (2.9)

Then the following implications hold.

(1) (i)⟹(ii)⟺(iii).

(2) (i)⟺(ii)⟺(iii) if X is both reflexive and smooth.

Proof. Note the following equivalence:

\begin{aligned}& J(x − x_{0}) ⋂ (N_{C}(x_{0}) + ∑_{i∈I} N_{C_{i}}(x_{0})) ≠ ∅ \\ ⟺ & J(x − x_{0})|_{C−x_{0}} ⋂ (N_{C}(x_{0})|_{C−x_{0}} + ∑_{i∈I} N_{C_{i}}(x_{0})|_{C−x_{0}}) ≠ ∅. \end{aligned}  (2.10)

Indeed, implication ⟹ in (2.10) 1s trivial; hence it suffices to show the converse implication. Thus, let x^{∗} ∈ J(x − x_{0}) be such that x^{∗}|_{C−x_{0}} ∈ J(x − x_{0})|_{C−x_{0}} ⋂ (N_{C}(x_{0})|_{C−x_{0}} + ∑_{i∈I} N_{C_{i}}(x_{0})|_{C−x_{0}}). Then there exist x_{0}^{∗} ∈ N_{C}(x_{0}), x_{i}^{∗} ∈ N_{C_{i}}(x_{0}), i = 1, 2, ⋯, m such that x^{∗}|_{C−x_{0}} = ∑_{i=0}^{m} x_{i}^{∗}|_{C−x_{0}}. Write y^{∗} = x^{∗} − ∑_{i=0}^{m} x_{i}^{∗}. Then y^{∗} ∈ N_{C}(x_{0}) and so x^{∗} = y^{∗} + ∑_{i=0}^{m} x_{i}^{∗} ∈ N_{C}(x_{0}) + ∑_{i∈I} N_{C_{i}}(x_{0}). Hence, x^{∗} ∈ J(x − x_{0}) ⋂ (N_{C}(x_{0}) + ∑_{i∈I} N_{C_{i}}(x_{0})). Therefore (2.10) is true.

Now, using (2.10), one can complete the proof in the same way as that given for [15, Theorem 3.1].

For the remainder of this section, let X denote a Hilbert space (over ℝ or ℂ), and we consider X^{∗} = X as usual. In particular, the normal cone of a nonempty set Z at z_{0} can be redefined N_{Z}(z_{0}) = {y ∈ X : Re ⟨y, z − z_{0}⟩ ≤ 0 for all z ∈ Z}. Let I(x_{0}) = {i ∈ I : x_{0} ∈ bd C_{i}}. Then, similar to the proof of [14, Theorem 4.1], we get the following theorem.

Theorem 2.4. Let X be a Hilbert space, K = C ∩ (∩_{i∈I}C_{i}) and let x_{0} ∈ K. Then the following statements are equivalent.

(i) The system {C, C_{i} : i ∈ I} has the strong CHIP at x_{0}.

(ii) For any x ∈ X, P_{K}(x) = x_{0} if and only if there exists a finite (possibly empty) set I_{0} ⊆ I such that P_{C}(x − ∑_{i∈I_{0}} h_{i}) = x_{0} for some h_{i} ∈ N_{C_{i}}(x_{0}) with each i ∈ I_{0}.

Now, let C be a closed convex set in X, {h_{i} : i ∈ I} ⊂ X \ {0} and let {Ω_{i} : i ∈ I} be a family of nonempty closed convex subsets of the scalar field. Define

Ĉ_{i} = {x ∈ X : ⟨x, h_{i}⟩ ∈ Ω_{i}}, i ∈ I,  (2.11)

and

K̂ = C ⋂ ( ⋂_{i∈I} Ĉ_{i}).  (2.12)

Let x_{0} ∈ K̂. For convenience, we shall write h̃_{i}(·) for the function ⟨h_{i}, ·⟩ on X, and h_{i}^{0} for the scalar ⟨h_{i}, x_{0}⟩. Then we have the following assertion:

NĈ_{i}(x_{0}) = \{\overline{α}h_{i} : α ∈ N_{Ω_{i}}(h_{i}^{0})\} for each i ∈ I.  (2.13)

This assertion was proved in the proof of [14, Theorem 4.2]. Here we give a direct and much simpler proof. In fact, it is direct that the set on the left-hand side contains the one on the right-hand side of (2.13). To show the converse inclusion, let h_{i}^{⊥} denote the orthogonal complement of h_{i} and let x^{∗} ∈ NĈ_{i}(x_{0}). Then, for each x ∈ h_{i}^{⊥} and γ ∈ ℂ, Re ⟨x^{∗}, γx⟩ ≤ 0 since γx + x_{0} ∈ Ĉ_{i}; hence x^{∗}⊥h_{i}^{⊥} and x^{∗} = αh_{i} for some scale α ∈ ℂ. Since, for each β ∈ Ω_{i}, there exists x ∈ Ĉ_{i} such that ⟨x^{∗}, x⟩ = β, we have that

Re \overline{α}(β − h_{i}^{0}) = ⟨x^{∗}, x − x_{0}⟩ ≤ 0.

This means that α ∈ N_{Ω_{i}}(h_{i}^{0}). Therefore x^{∗} belongs to the set on the right-hand side of (2.13) and (2.13) is proved. Thus, by (2.13) and Theorem 2.4, we immediately obtain the following perturbation theorem, which was given in [14]. Note that the proof here is much simpler than that in [14].

Corollary 2.1. Let X be a Hilbert space and let x_{0} ∈ K̂, where K̂ is defined by (2.12). Then the following statements are equivalent.

(i) The collection of closed convex sets {C, Ĉ_{i} : i ∈ I} has the strong CHIP at x_{0}.

(ii) For any x ∈ X, PK̂(x) = x_{0} if and only if there exists a finite (possibly empty) set I_{0} ⊆ I such that P_{C}(x − ∑_{i∈I_{0}} \overline{α}_{i}h_{i}) = x_{0} for some α_{i} ∈ N_{Ω_{i}}(h_{i}^{0}) with each i ∈ I_{0}.

3 Characterization for constrained approximation in complexvalued function spaces

Let C(Q) denote the Banach space of all complex-valued continuous functions on a compact metric space Q endowed with the uniform norm:

∥f∥ = max_{t∈Q} |f(t)| for each f ∈ C(Q).  (3.1)

Let P be a n-dimensional subspace of C(Q) and {Ω_{t} : t ∈ Q} a family of nonempty closed convex sets in the complex plane ℂ. For brevity, we write {Ω_{t}} for {Ω_{t} : t ∈ Q}. Note that, for each t ∈ Q, Ω_{t} is either a point, or a linear-segment or a “planar” convex set (of real dimension 2) in the complex plane ℂ. Set

P_{Ω} = {p ∈ P : p(t) ∈ Ω_{t} for each t ∈ Q}.  (3.2)

The problem considered here is that of finding an element p^{∗} ∈ P_{Ω} for a function f ∈ C(Q) such that

∥f − p^{∗}∥ = inf_{p∈P_{Ω}} ∥f − p∥,  (3.3)

(such p^{∗} is called a best restricted range approximation to f from P with respect to {Ω_{t}}, see [24, 28, 11, 14]).

We assume that

Q = Q_{S} ⋃ Q_{E} ⋃ Q_{N},  (3.4)

where

Q_{S} = {t ∈ Q : Ω_{t} is a singleton},
Q_{E} = {t ∈ Q \ Q_{S} : intΩ_{t} = ∅},
Q_{N} = {t ∈ Q : intΩ_{t} ≠ ∅}.

We also assume for the whole section that

Q_{S} ∪ Q_{E} is finite.  (3.5)

We introduce some short notation of conditions for easy reference.

• IC_{0}: there exists an element p̄ ∈ P_{Ω} such that p̄(t) ∈ int Ω_{t} for each t ∈ Q, that is,

0 ∈ ⋂_{t∈Q} int (Ω_{t} − p̄(t)).  (3.6)

• IC: there exists an element p̄ ∈ P_{Ω} such that

0 ∈ int ( ⋂_{t∈Q_{N}}(Ω_{t} − p̄(t))) ⋂ ( ⋂_{t∈Q_{E}} ri(Ω_{t} − p̄(t)));  (3.7)

• UKC: the set-valued function t ↦ Ω_{t} is upper Kuratowski semicontinuous on Q;

• LKC: the set-valued function t ↦ Ω_{t} is lower Kuratowski semicontinuous on Q;

• KC: the set-valued function t ↦ Ω_{t} is Kuratowski continuous on Q.

We will see latter that these conditions closely relate to some corresponding properties of the CCS-system {P, C_{t} : t ∈ Q} in C(Q), where C_{t} is defined by (1.3). Let f ∈ C(Q) and p^{∗} ∈ P_{Ω}. We fix this pair of function f, p^{∗} in what follows. Define

σ(t) = f(t) − p^{∗}(t) for each t ∈ Q.  (3.8)

Set

M(σ) = {t ∈ Q : |σ(t)| = ∥σ∥}

and

B(p^{∗}) = {t ∈ Q : p^{∗}(t) ∈ bdΩ_{t}}, B^{rb}(p^{∗}) = {t ∈ Q \ Q_{S} : p^{∗}(t) ∈ rbΩ_{t}}.

(Here we adopt the convention that bd Ω_{t} = Ω_{t} if Ω_{t} is a singleton.) Note that

B^{rb}(p^{∗}) = (B(p^{∗}) ∩ Q_{N}) ∪ {t ∈ Q_{E} : p^{∗}(t) ∈ rbΩ_{t}}  (3.9)

and in particular that B^{rb}(p^{∗}) ⊆ B(p^{∗}). Moreover B^{rb}(p^{∗}) = B(p^{∗}) in the case when Q_{S} and Q_{E} are empty (e.g., when IC_{0} holds).

Let span_{R} (Ω_{t} − p^{∗}(t)) denote the real subspace spanned by Ω_{t}−p^{∗}(t) in ℂ. Then span_{R} (Ω_{t} − p^{∗}(t)) is the whole complex plane ℂ if t ∈ Q_{N}, a line in ℂ if t ∈ Q_{E} and the singleton {0} if t ∈ Q_{S}. Set

P_{R} = {p ∈ P : p(t) ∈ span_{R} (Ω_{t} − p^{∗}(t)) for each t ∈ Q_{E} ∪ Q_{S}}.  (3.10)

Note that P_{R} is a real subspace of P and that P_{R} = P if Q = Q_{N}. Let m denote the real dimension of P_{R}: dim_{R}P_{R} = m, and let ψ_{1}, ⋯, ψ_{m} be a real basis of P_{R}, that is, each element of P_{R} can be uniquely expressed as a real linear combination of ψ_{1}, ⋯, ψ_{m}. Moreover, let {ϕ_{1}, ⋯, ϕ_{n}} be a (complex) basis of P, that is, each element of P can be uniquely expressed as a complex linear combination of ϕ_{1}, ⋯, ϕ_{n}.

We define

τ(t) = {τ ∈ −N_{Ω_{t}}(p^{∗}(t)) : |τ| = 1} for each t ∈ Q.  (3.11)

Note that if t ∈ Q_{N} ∩ B(p^{∗}) and τ ∈ τ(t) then

Re \overline{τ}(z − p^{∗}(t)) > 0  (3.12)

for all z ∈ int Ω_{t}. Since int Ω_{t} = ∅ if t ∈ Q \ Q_{N}, we are led to define two more set-valued functions from Q to the unit sphere of ℂ:

τ_{r}(t) = \{ \begin{matrix}τ(t) & for each t ∈ Q \ Q_{E}, \\ \{τ ∈ ℂ : |τ| = 1, Re \overline{τ}(z − p^{∗}(t)) > 0 ∀z ∈ ri Ω_{t}\} & for each t ∈ Q_{E}. \end{matrix} ,  (3.13)

and

τ_{r}^{+}(t) = \{ \begin{matrix}τ(t) & for each t ∈ Q \ Q_{E}, \\ ∅ & for each t ∈ Q_{E} with p^{∗}(t) ∈ ri Ω_{t}, \\ \frac{z−p^{∗}(t)}{|z−p^{∗}(t)|} & for each t ∈ Q_{E} with p^{∗}(t) ∈ rb Ω_{t}, z ∈ Ω_{t} \ p^{∗}(t). \end{matrix}  (3.14)

(Note that \frac{z−p^{∗}(t)}{|z−p^{∗}(t)|} does not depend on the particular choice of z as Ω_{t} is a line-segment for t ∈ Q_{E}.)

Remark 3.1. (i) For any t ∈ Q, τ(t) ≠ ∅ ⟺ t ∈ B(p^{∗}).

(ii) For any t ∈ Q_{E},

τ_{r}(t) ≠ ∅ ⟺ t ∈ B^{rb}(p^{∗}) ⟺ τ_{r}^{+}(t) is a singleton .  (3.15)

(iii) If t ∈ B^{rb}(p^{∗}) ∩ Q_{E} and τ ∈ −N_{Ω_{t}}(p^{∗}(t)) with |τ| = 1, then

τ ∉ τ_{r}(t) ⟺ Re \overline{τ}(z −p^{∗}(t)) = 0 for each z ∈ Ω_{t} ⟺ Re \overline{τ}(z −p^{∗}(t)) = 0 for some z ∈ Ω_{t}. (3.16)

(iv) For any t ∈ Q, τ_{r}^{+}(t) is compact

τ_{r}^{+}(t) ⊆ τ_{r}(t) ⊆ τ(t).  (3.17)

Let t ∈ B^{rb}(p^{∗}) ∩ Q_{E}, τ ∈ τ_{r}(t) and let Pr_{t}(τ) denote the projection of τ on the subspace span_{R} (Ω_{t} − p^{∗}(t)). Then Pr_{t}(τ) ≠ 0,

\frac{Pr_{t}(τ)}{|Pr_{t}(τ)|} ∈ τ_{r}^{+}(t) and Re z\overline{τ} = Re z\overline{Pr_{t}(τ)} for each z ∈ span_{R} (Ω_{t} − p^{∗}(t)) .  (3.18)

For each t ∈ Q, let c(t) ⊂ ℂ^{n}, c_{r}(t) ⊂ ℝ^{m} and c_{r}^{+}(t) be defined respectively by

c(t) := \{(ϕ_{1}(t)\overline{τ}, ⋯, ϕ_{n}(t)\overline{τ}) : τ ∈ τ(t)\},  (3.19)
c_{r}(t) := \{(Re ψ_{1}(t)\overline{τ}, ⋯, Re ψ_{m}(t)\overline{τ}) : τ ∈ τ_{r}(t)\}.  (3.20)

and

c_{r}^{+}(t) := \{(Re ψ_{1}(t)\overline{τ}, ⋯, Re ψ_{m}(t)\overline{τ}) : τ ∈ τ_{r}^{+}(t)\}.  (3.21)

Set

U = ⋃_{t∈B(p^{∗})} c(t),  U_{r} = ⋃_{t∈B^{rb}(p^{∗})} c_{r}(t),  U_{r}^{+} = ⋃_{t∈B^{rb}(p^{∗})} c_{r}^{+}(t).  (3.22)

Note that these sets are bounded and that, by (3.17) and (3.18),

U_{r}^{+} ⊆ U_{r} ⊆ ⋃_{0<η≤1} (η U_{r}^{+}).  (3.23)

Recalling (3.8), we define b(t) ∈ ℂ^{n} and b_{r}(t) ∈ ℝ^{m} respectively by

b(t) = (ϕ_{1}(t), ⋯, ϕ_{n}(t))\overline{σ(t)} = (ϕ_{1}(t)\overline{σ(t)}, ⋯, ϕ_{n}(t)\overline{σ(t)}) for each t ∈ Q  (3.24)

and

b_{r}(t) = Re (ψ_{1}(t), ⋯, ψ_{m}(t))\overline{σ(t)} for each t ∈ Q.  (3.25)

We define

V = {b(t) : t ∈ M(σ)},  V_{r} = {b_{r}(t) : t ∈ M(σ)}.  (3.26)

Clearly they are compact sets. Set

W = V ⋃ U,  W_{r} = V_{r} ⋃ U_{r},  W_{r}^{+} = V_{r} ⋃ U_{r}^{+}.  (3.27)

Note that they are bounded sets. Also, by (3.23),

W_{r}^{+} ⊆ W_{r} ⊆ ⋃_{0<t≤1} (t W_{r}^{+}).  (3.28)

This implies that

co W_{r}^{+} ⊆ co W_{r} ⊆ co ⋃_{0<t≤1} (t W_{r}^{+}) ⊆ ⋃_{0<t≤1} (t co W_{r}^{+}),  (3.29)

where the last inclusion can be verified by a routine verification.

Let

C_{t} = {u ∈ C(Q) : u(t) ∈ Ω_{t}} for each t ∈ Q.  (3.30)

Then

P_{Ω} = P ⋂( ⋂_{t∈Q} C_{t}).  (3.31)

Clearly {P, C_{t} : t ∈ Q} is a CCS-system with base-set P. To prepare for our main result, we first give a few lemmas. These lemmas will show in particular that the conditions introduced at the beginning of this section for the system {Ω_{t}} are naturally linked to some desirable properties of the system {P, C_{t} : t ∈ Q} so that the results in Sections 2 are applicable. The first of the lemmas describes the connections of the conditions IC_{0}, IC for the system {Ω_{t}} and the interior-point conditions for the system {P, C_{t} : t ∈ Q} while the second the connection of the normal cones of Ω_{t} and to that of the corresponding C_{t}.

Lemma 3.1. (i) The system {Ω_{t}} satisfies IC_{0} if and only if the CCS-system {P, C_{t} : t ∈ Q} satisfies the interior-point condition. Furthermore, 0 ∉ conv U if the system {Ω_{t}} satisfies IC_{0}.

(ii) The system {Ω_{t}} satisfies IC if and only if the CCS-system {P, C_{t} : t ∈ Q} satisfies the weak-strong interior-point condition with the pair (Q_{E}, Q_{S}). Furthermore, 0 ∉ conv U_{r} if the system {Ω_{t}} satisfies IC.

Proof. Let α > 0 and f_{0} ∈ C_{t}. We claim that

B(f_{0}, α) ⊆ C_{t} ⟺ B(f_{0}(t), α) ⊆ Ω_{t} for each t ∈ Q_{N},  (3.32)
B(f_{0}, α) ⋂ aff C_{t} ⊆ C_{t} ⟺ B(f_{0}(t), α) ⋂ aff Ω_{t} ⊆ Ω_{t} for each t ∈ Q_{E}.  (3.33)

We shall only prove (3.33) (the proof of (3.32) is similar). To do this, we need only to establish the necessity part. Note first the following obvious fact:

aff C_{t} = {u ∈ C(Q) : u(t) ∈ aff Ω_{t}} for each t ∈ Q.  (3.34)

Let t ∈ Q_{E} and assume that

B(f_{0}, α) ⋂ aff C_{t} ⊆ C_{t}.  (3.35)

Let z ∈ B(f_{0}(t), α) ⋂ aff Ω_{t}. We have to show that z ∈ Ω_{t}. By the Tietze Extension Theorem, there exists s ∈ C(Q) such that ∥s∥ = s(t) = 1. Define

f(w) = f_{0}(w) + s(w)(z − f_{0}(t)) ∀w ∈ Q.

Then ∥f − f_{0}∥ ≤ |z − f_{0}(t)| ≤ α. Since f(t) = z ∈ aff Ω_{t}, f ∈ aff C_{t} by (3.34). Consequently, f ∈ C_{t} and hence z = f(t) ∈ Ω_{t}, as required. Therefore our claim stands.

By (3.32), we have that

int C_{t} = {u ∈ C(Q) : u(t) ∈ int Ω_{t}} for each t ∈ Q.  (3.36)

Thus the first part of (i) is clear. Again by (3.32),

int ⋂_{t∈Q_{N}} C_{t} = {u ∈ C(Q) : u(t) ∈ int ⋂_{t∈Q_{N}} Ω_{t}},  (3.37)

while, by (3.33),

ri C_{t} = {u ∈ C(Q) : u(t) ∈ ri Ω_{t}} for each t ∈ Q_{E}.  (3.38)

Combining (3.37) and (3.38), the first part of (ii) is also clear.

Thus, to complete the proof, it remains to show that (a): 0 ∉ conv U_{r} if IC is satisfied and that (b): 0 ∉ conv U if IC_{0} is satisfied. We shall only prove (a) as the proof for (b) is similar. Suppose that there exist λ_{1}, ⋯, λ_{s} ∈ [0, 1] with ∑_{j=1}^{s} λ_{j} = 1 and t′_{1}, ⋯, t′_{s} ∈ B^{rb}(p^{∗}), τ′_{j} ∈ τ_{r}(t′_{j}), j = 1, ⋯, s such that

Re ∑_{j=1}^{s} p(t′_{j})λ_{j}\overline{τ′_{j}} = 0  (3.39)

holds for each p ∈ {ψ_{1}, ⋯, ψ_{m}} and hence for each p ∈ P_{R}. Assuming IC with some p̄ ∈ P_{Ω} satisfying (3.7), let p := p̄− p^{∗}. Since, each t′_{j} ∈ B^{rb}(p^{∗}) and each τ′_{j} ∈ τ_{r}(t′_{j}), we get, by (3.12), (3.7) and (3.13) that

Re p(t′_{j})\overline{τ}′_{j} = Re (p̄(t′_{j}) − p^{∗}(t′_{j}))\overline{τ}′_{j} > 0 for each j = 1, ⋯, s.  (3.40)

This contradicts (3.39) and hence 0 ∉ conv U_{r}.

Lemma 3.2. Let t ∈ Q and assume that p^{∗} ∈ C_{t}. Then

N_{C_{t}}(p^{∗}) = \{\overline{α}e_{t} : α ∈ N_{Ω_{t}}(p^{∗}(t))\},  (3.41)

where e_{t} denotes the point-functional on C(Q) defined by

⟨e_{t}, u⟩ = u(t) for each u ∈ C(Q).  (3.42)

Proof. Let u ∈ C(Q). Let z ∈ Ω_{t} be such that d_{Ω_{t}}(u(t)) = |z − u(t)|. By the Tietze Extension Theorem, there exists a function w ∈ C(Q) such that ∥w∥ = |u(t) − z| and w(t) = u(t) − z (so u − w ∈ C_{t}). Then d_{C_{t}}(u) ≤ ∥u − (u − w)∥ = |z − u(t)| = d_{Ω_{t}}(u(t)). Consequently,

d_{C_{t}}(u) = d_{Ω_{t}}(u(t)) for each u ∈ C(Q).  (3.43)

as it is straightword to verify that d_{C_{t}}(u) ≥ d_{Ω_{t}}(u(t)). Since p^{∗} ∈ C_{t} (and so p^{∗}(t) ∈ Ω_{t}), (3.43) and the proof of [14, Lemma 5.2 (iii)] imply that

∂d_{C_{t}}(p^{∗}) = \{\overline{α}e_{t} ∈ C(Q)^{∗} : α ∈ ∂d_{Ω_{t}}(p^{∗}(t))\}.  (3.44)

Recalling from [5] that

∂d_{C_{t}}(p^{∗}) = {x^{∗} ∈ N_{C_{t}}(p^{∗}) : ∥x^{∗}∥ ≤ 1} and ∂d_{Ω_{t}}(p^{∗}(t)) = {α ∈ N_{Ω_{t}}(p^{∗}(t)) : |α| ≤ 1}, (3.45)

it follows that (3.41) holds.

Lemma 3.3. (i) If UKC is satisfied, then the set-valued function t ↦ C_{t} is upper Kuratowski semicontinuous on Q.

(ii) If LKC is satisfied, then the set-valued function t ↦ C_{t} is lower Kuratowski semicontinuous on Q.

Proof. Let t_{0} ∈ Q and {t_{k}} ⊆ Q be a sequence convergent t_{0}.

(i) Let f_{k} ∈ C_{t_{k}} for each k be such that ∥f_{k} − f̄∥ → 0. Then, f_{k}(t_{k}) ∈ Ω_{t_{k}} for each k and f_{k}(t_{k}) → f̄(t_{0}) as k → ∞. By the condition UKC, it follows that f̄(t_{0}) ∈ Ω_{t_{0}} and so f̄ ∈ C_{t_{0}}. This proves (i).

(ii) Let f_{0} ∈ C_{t_{0}}. Then f_{0}(t_{0}) ∈ Ω_{t_{0}} and, by the condition LKC, there exists z_{k} ∈ Ω_{t_{k}} for each k such that |z_{k} − f_{0}(t_{0})| → 0. Define f_{k} ∈ C(Q) by

f_{k}(t) = f_{0}(t) + z_{k} − f_{0}(t_{k}) for each t ∈ Q.

Thus f_{k}(t_{k}) = z_{k} ∈ Ω_{t_{k}} and hence f_{k} ∈ C_{t_{k}}. Moreover, we also have that

∥f_{k} − f_{0}∥ = |z_{k} − f_{0}(t_{k})| ≤ |z_{k} − f_{0}(t_{0})| + |f_{0}(t_{0}) − f_{0}(t_{k})| → 0.

Thus (ii) is proved.

Lemma 3.4. Suppose that the condition LKC is satisfied. Then B(p^{∗}) is closed and W is compact in ℂ^{n}.

Proof. Let {t_{k}} ⊆ B(p^{∗}) and {τ_{k}} ⊆ ∪_{t∈B(p^{∗})}τ(t) be such that τ_{k} ∈ τ(t_{k}), t_{k} → t_{0} ∈ Q and τ_{k} → τ ∈ ℂ. Then |τ_{k}| = |τ| = 1. Moreover, since Q \ Q_{N} is finite, we assume, without loss of generality, that each t_{k} ∈ Q_{N}. Then, for each k,

Re \overline{−τ_{k}}(z − p^{∗}(t_{k})) ≤ 0 for each z ∈ Ω_{t_{k}}.  (3.46)

By the condition LKC, for each z ∈ Ω_{t_{0}}, there exists z_{k} ∈ Ω_{t_{k}} such that z_{k} → z. Noting that p^{∗}(t_{k}) → p^{∗}(t_{0}), it follows from (3.46) that

Re \overline{−τ}(z − p^{∗}(t_{0})) ≤ 0 for all z ∈ Ω_{t_{0}}.  (3.47)

Since p^{∗}(t_{0}) ∈ Ω_{t_{0}} as p^{∗} ∈ P_{Ω}, this means that −τ ∈ N_{Ω_{t_{0}}} (p^{∗}(t_{0})). Since τ ≠ 0, this implies that p^{∗}(t_{0}) ∈ bdΩ_{t_{0}} and so t_{0} ∈ B(p^{∗}(t_{0})). Hence, B(p^{∗}) is closed and hence τ ∈ ∪_{t∈B(p^{∗})}τ(t). This shows that ∪_{t∈B(p^{∗})}τ(t) is closed and hence compact since it is bounded. By definition, it is now easily verified that U is compact. Since V is compact, it follows that W is compact.

Lemma 3.5. Suppose that the conditions LKC and IC hold. Then B^{rb}(p^{∗}) is closed and the closure of W_{r}^{+} is contained in W_{r}.

Proof. As in the proof of Lemma 3.4, let {t_{k}} ⊆ B^{rb}(p^{∗}), τ_{k} ∈ τ_{r}^{+}(t_{k}) for each k such that t_{k} → t_{0} ∈ Q and τ_{k} → τ ∈ ℂ. Thus, by (3.9) and (3.17), one has {t_{k}} ⊆ B(p^{∗}), τ_{k} ∈ τ(t_{k}) for each k. By Lemma 3.4, it follow that t_{0} ∈ B(p^{∗}) and −τ ∈ N_{Ω_{t_{0}}} (p^{∗}(t_{0})) thanks to LKC. It suffices to show that t_{0} ∈ B^{rb}(p^{∗}) and τ ∈ τ_{r}(t_{0}). If t_{0} ∈ Q_{N}, they are done by the proof of Lemma 3.4 because one has then t_{0} ∈ B(p^{∗}) ∩ Q_{N} ⊆ B^{rb}(p^{∗}) and τ ∈ τ(t_{0}) = τ_{r}(t_{0}). Thus we may assume henceforth that t_{0} ∉ Q_{N}. Note that if t_{k} ∈ Q_{E} for infinitely many k, then, since Q_{E} is finite, one has t_{k} = t_{0} for these k (say for all k by considering a subsequence if necessary). Hence t_{0} ∈ B^{rb}(p^{∗}) and τ_{k} ∈ τ_{r}(t_{0}). However, in view of (3.15), τ_{r}(t_{0}) must be a singleton in the present case, so τ ∈ τ_{r}(t_{0}). Therefore, without loss of generality, we may assume that t_{k} ∈ Q_{N} for each k. In view of (3.27), to complete the proof, it is sufficient to show that t_{0} ∈ Q_{E}, p^{∗}(t_{0}) ∈ rb Ω_{t_{0}} and

Re \overline{τ}(z − p^{∗}(t_{0})) > 0 for some z ∈ ri Ω_{t_{0}}.  (3.48)

By IC, there exists p̄ ∈ P_{Ω} satisfying (3.7). Let δ > 0 be such that

B(0, δ) ⊂ ⋂_{t∈Q_{N}} (Ω_{t} − p̄(t)) .  (3.49)

We will show that there exists an integer N > 0 such that

B(p̄(t_{0}) − p^{∗}(t_{0}), \frac{δ}{2}) ⊂ ⋂_{k≥N} (Ω_{t_{k}} − p^{∗}(t_{k})) .  (3.50)

Indeed, take N > 0 such that |(p̄(t_{k}) − p^{∗}(t_{k})) − (p̄(t_{0}) − p^{∗}(t_{0}))| < \frac{δ}{2} for each k ≥ N. Then

B(p̄(t_{0}) − p^{∗}(t_{0}), \frac{δ}{2}) ⊂ B(p̄(t_{k}) − p^{∗}(t_{k}), δ) for each k ≥ N.  (3.51)

On the other hand, by (3.49),

B(p̄(t_{k}) − p^{∗}(t_{k}), δ) ⊂ Ω_{t_{k}} − p^{∗}(t_{k}) for each k.  (3.52)

Consequently, (3.50) follows from (3.51) and (3.52). Set Ω^{∗} := ⋂_{k≥N} (Ω_{t_{k}} − p^{∗}(t_{k})). Then 0 ∈ bd Ω^{∗} and p̄(t_{0}) − p^{∗}(t_{0}) ∈ int Ω^{∗} by (3.50). In particular,

Re \overline{α}(p̄(t_{0}) − p^{∗}(t_{0})) < 0 for each α ∈ N_{Ω^{∗}}(0) \ \{0\}.

Hence, there exists positive number b such that, for each α ∈ N_{Ω^{∗}}(0) with |α| = 1,

Re \overline{α}(p̄(t_{0}) − p^{∗}(t_{0})) ≤ −b < 0.  (3.53)

Since −τ_{k} ∈ N_{Ω_{t_{k}}} (p^{∗}(t_{k})), | − τ_{k}| = 1 and N_{Ω_{t_{k}}} (p^{∗}(t_{k})) ⊆ N_{Ω^{∗}}(0) for each n ≥ N, we have that

Re \overline{−τ_{k}}(p̄(t_{0}) − p^{∗}(t_{0})) ≤ −b < 0 for each k ≥ N.  (3.54)

Noting that τ_{k} → τ, it follows that

Re \overline{−τ}(p̄(t_{0}) − p^{∗}(t_{0})) ≤ −b < 0.  (3.55)

Thus Ω_{t_{0}} contains more than one point (p̄(t_{0}), p^{∗}(t_{0}) being distinct members of Ω_{t_{0}}). It follows that Ω_{t_{0}} is a line-segment (recalling that t_{0} ∉ Q_{N}), i.e., t_{0} ∈ Q_{E}. Consequently, by (3.7), p̄(t_{0}) ∈ ri Ω_{t_{0}}. Therefore (3.48) holds by (3.55). Since 0 ≠ −τ ∈ N_{Ω_{t_{0}}} (p^{∗}(t_{0})) (noting p̄(t_{0}) ∈ Ω_{t_{0}}), it follows from (3.55) that p^{∗}(t_{0}) must be an end point of Ω_{t_{0}}, i.e., p^{∗}(t_{0}) ∈ rb Ω_{t_{0}}. The proof of Lemma 3.5 is complete.

Lemma 3.6. Let Φ be a complex linear functional on P such that

Re Φ(p) = 0 for each p ∈ P_{R}.  (3.56)

Then there exist a nonnegative integer s with s ≤ 2n − m, {t′′_{j}}_{j=1}^{s} ⊆ Q_{E} ∪ Q_{S}, {τ′′_{j}}_{j=1}^{s} ⊂ ℂ \ {0} with each τ′′_{j} ∈ −N_{Ω_{t{′′ \atop j}}} (p^{∗}(t′′_{j})) such that

Φ(p) + ∑_{j=1}^{s} p(t′′_{j})\overline{τ′′_{j}} = 0 for each p ∈ P.  (3.57)

Proof. We assume that Q_{E} ∪ Q_{S} ≠ ∅ (the result is trivial otherwise). For each t ∈ Q_{E}, span_{R}(Ω_{t} − p^{∗}(t)) is a line passing through the origin. Hence there exists τ_{t} ∈ ℂ with |τ_{t}| = 1 which is “perpendicular” to span_{R}(Ω_{t} − p^{∗}(t)) in the sense that

Re \overline{τ_{t}} α = 0 ⟺ α ∈ span_{R}(Ω_{t} − p^{∗}(t)).  (3.58)

Thus, defining the real linear functional ξ_{t} on P by

ξ_{t}(p) = Re \overline{τ_{t}} p(t), for each p ∈ P,  (3.59)

we can characterize the kernel of ξ_{t} for t ∈ Q_{E}:

p ∈ Ker ξ_{t} ⟺ p(t) ∈ span_{R}(Ω_{t} − p^{∗}(t)).  (3.60)

For each t ∈ Q_{S}, two linear functionals on P (respectively denoted by ξ_{t} and ξ′_{t}) will be useful for us. They are defined by

ξ_{t}(p) = Re p(t), for each p ∈ P,

ξ′_{t}(p) = Re ip(t), for each p ∈ P,

where i = \sqrt{−1}. Thus, for t ∈ Q_{S},

p(t) = 0 ⟺ p ∈ Ker ξ_{t} ⋂ Ker ξ′_{t}.

By (3.10), we have that

P_{R} = ( ⋂_{t∈Q_{E}∪Q_{S}} Ker ξ_{t}) ⋂ ( ⋂_{t∈Q_{S}} Ker ξ′_{t}) .  (3.61)

It will be convenient to list the functionals

{ξ_{t} : t ∈ Q_{E} ∪ Q_{S}} ⋃{ξ′_{t} : t ∈ Q_{S}} := {ξ^{1}, ξ^{2}, ⋯, ξ^{r}},  (3.62)

where r = |Q_{E}| + 2|Q_{S}|, where for example |Q_{E}| stands for the number of elements in Q_{E}. Letting q := 2n−m, the difference of real dimensions of P and P_{R}, one has q ≤ r. Recalling that {ψ_{1}, ⋯, ψ_{m}} is a basis of P_{R}, there exist ψ_{m+1}, ⋯, ψ_{2n} ∈ P such that {ψ_{1}, ⋯, ψ_{2n}} is a real basis of P. Since P_{R}∩span_{R}\{ψ_{m+1}, ⋯, ψ_{m+q}\} = \{0\}, it is easy to verify that the vectors \{\overrightarrow{a}_{i} : i = m+1, ⋯, m+q\} ⊂ ℝ^{r} are (real) linearly independent, where each \overrightarrow{a}_{i} is defined by

\overrightarrow{a}_{i} = (ξ^{ν}(ψ_{i}))_{ν=1}^{r} ∈ ℝ^{r} for each i = m + 1, ⋯, m + q.

Consequently, there exist q-many coordinates such that the restrictions \overrightarrow{a}_{i}| of \overrightarrow{a}_{i} (m+1 ≤ i ≤ m+q) to these coordinates are linearly independent. Without loss of generality, let us assume that they are the first q coordinates; thus

det (ξ^{ν}(ψ_{i}))_{1≤ν≤q, m+1≤m+q} ≠ 0.  (3.63)

Therefore there exist real numbers (λ′_{1}, ⋯, λ′_{q}) such that

∑_{ν=1}^{q} λ′_{ν}ξ^{ν}(ψ_{i}) = −Re Φ(ψ_{i}) for i = m + 1, ⋯, m + q.  (3.64)

Note that, for i = 1, 2, ⋯, m, both sides of (3.64) are equal to zero thanks to (3.56) and (3.61). Therefore

∑_{ν=1}^{q} λ′_{ν}ξ^{ν}(p) + Re Φ(p) = 0  (3.65)

for each p ∈ {ψ_{1}, ⋯, ψ_{m}, ψ_{m+1}, ⋯, ψ_{m+q}}. In view of (3.65), it is clear that, to complete the proof, it suffices to show that the first summation in (3.65) can be expressed in the form

∑_{ν=1}^{q} λ′_{ν}ξ^{ν}(p) = Re ∑_{j=1}^{s} p(t′′_{j})\overline{τ′′_{j}} for each p ∈ P  (3.66)

for some s ≤ q, {t′′_{j}}_{j=1}^{s} ⊆ Q_{E} ∪ Q_{S}, {τ′′_{j}}_{j=1}^{s} ⊂ ℂ \ {0} such that

τ′′_{j} ∈ −N_{Ω_{t{′′ \atop j}}} (p^{∗}(t′′_{j})) for each j = 1, 2, ⋯, s.  (3.67)

To do this, we consider, in light of (3.62), ν with 1 ≤ ν ≤ q for each of the following cases. (a) ξ^{ν} = ξ_{t} for some t ∈ Q_{E}. Then τ′′_{t} := λ′_{ν}τ_{t} ∈ −N_{Ω_{t}}(p^{∗}(t)) by (3.58), and by (3.59),

(λ′_{ν}ξ^{ν})(p) = λ′_{ν} Re \overline{τ_{t}}p(t) = Re p(t)\overline{τ′′_{t}} for each p ∈ P.

(b) ξ^{ν} = ξ_{t} for some t ∈ Q_{S} but ξ′_{t} ∉ {ξ^{1}, ξ^{2}, ⋯, ξ^{r}}. Then τ′′_{t} := λ′_{ν} ∈ −N_{Ω_{t}}(p^{∗}(t)) as Ω_{t} = {p^{∗}(t)}, and

(λ′_{ν}ξ^{ν})(p) = λ′_{ν} Re p(t) = Re p(t)\overline{τ′′_{t}} for each p ∈ P.

(c) ξ^{ν} = ξ′_{t} for some t ∈ Q_{S} but ξ_{t} ∉ \{ξ^{1}, ξ^{2}, ⋯, ξ^{r}\}. Then τ′′_{t} := \overline{iλ′_{ν}} ∈ −N_{Ω_{t}}(p^{∗}(t)) and

(λ′_{ν}ξ^{ν})(p) = λ′_{ν} Re ip(t) = Re p(t)\overline{τ′′_{t}} for each p ∈ P.

(d) ξ^{ν} = ξ_{t} for some t ∈ Q_{S} which satisfies an additional property that ξ′_{t} ∈ {ξ^{1}, ξ^{2}, ⋯, ξ^{r}}. Assume that ξ′_{t} = ξ^{ν′}. Then τ′′_{t} := λ′_{ν} + \overline{iλ′_{ν′}} ∈ −N_{Ω_{t}}(p^{∗}(t)) as Ω_{t} = \{p^{∗}(t)\}, and

λ′_{ν}ξ^{ν}(p) + λ′_{ν′}ξ^{ν′}(p) = λ′_{ν} Re p(t) + λ′_{ν′}Re \overline{i} p(t) = Re p(t)\overline{τ′′_{t}} for each p ∈ P.

Combining (a-d) and deleting these terms with the corresponding τ′′_{t} = 0, (3.66) is seen to hold.

In the following Theorems 3.1-3.5, we consider relations of the following statements for a fixed pair of functions f ∈ C(Q) and p^{∗} ∈ P_{Ω}. Recall that σ := f − p^{∗}.

(i) p^{∗} is a best restricted range approximation to f from P with respect to {Ω_{t}}. (ii) For each p ∈ P_{R}, there exist t ∈ M(σ), t′ ∈ B^{rb}(p^{∗}) such that

max\{Re (p(t)\overline{σ(t)}), max_{τ∈τ_{r}^{+}(t′)} Re (p(t′)\overline{τ})\} ≥ 0.  (3.68)

(iii) For each p ∈ P_{R}, there exist t ∈ M(σ), t′ ∈ B^{rb}(p^{∗}) and τ ∈ τ_{r}(t′) such that

max\{Re (p(t)\overline{σ(t)}), Re (p(t′)\overline{τ})\} ≥ 0.  (3.69)

(iv) The origin of ℝ^{m} belongs to the convex hull of the W_{r}^{+}.

(v) The origin of ℝ^{m} belongs to the convex hull of the W_{r}. (vi) The origin of ℂ^{n} belongs to the convex hull of the W. (vii) There exist

{t_{i}}_{i=1}^{k} ⊆ M(σ), {λ_{i}}_{i=1}^{k} ⊂ (0, +∞)

and

{t′_{j}}_{j=1}^{l} ⊆ B^{rb}(p^{∗}), {τ′_{j}}_{j=1}^{l} ⊂ ℂ \ {0}

with 1 + l ≤ k + l ≤ m + 1 such that τ′_{j} ∈ −N_{Ω_{t′_{j}}} (p^{∗}(t′_{j})) for each j = 1, ⋯, l, and

Re ∑_{i=1}^{k} λ_{i}p(t_{i})\overline{σ(t_{i})} + Re ∑_{j=1}^{l} p(t′_{j})\overline{τ′_{j}} = 0 for each p ∈ P_{R}.  (3.70)

(viii) There exist

{t_{i}}_{i=1}^{k} ⊆ M(σ), {λ_{i}}_{i=1}^{k} ⊂ (0, +∞)  (3.71)

and

{t′_{j}}_{j=1}^{l′} ⊆ B(p^{∗}), {τ′_{j}}_{j=1}^{l′} ⊂ ℂ \ {0}  (3.72)

with 1 + l′ ≤ k + l′ ≤ 2n + 1 such that τ′_{j} ∈ −N_{Ω_{t′_{j}}} (p^{∗}(t′_{j})) for each j = 1, ⋯, l′, and

∑_{i=1}^{k} λ_{i}p(t_{i})\overline{σ(t_{i})} + ∑_{j=1}^{l′} p(t′_{j})\overline{τ′_{j}} = 0 for each p ∈ P.  (3.73)

(ix) J(σ)|_{P} ⋂ (∑_{t∈Q} N_{C_{t}}(p^{∗})|_{P}) ≠ ∅.

Theorem 3.1. The following implications hold.

(vii) ⟺ (viii) ⟺ (ix) ⟹ (iv) ⟹ (ii) ⟺ (iii)
⇓  ⇕
(i)  (v) ⟹ (vi)

Proof. By (3.29), it is easy to verify that (iv)⟺(v). Also by (3.17) and (3.18), we have (ii)⟺(iii). Applying Lemma 3.6 to the functional Φ on P defined by

Φ(p) = ∑_{i=1}^{k} λ_{i}p(t_{i})\overline{σ(t_{i})} + Re ∑_{j=1}^{l} p(t′_{j})\overline{τ′_{j}} for each p ∈ P,

we have that (vii)⟹(viii) with l′ = l + s, where s is as in Lemma 3.6. To show (viii)⟹ (vii)⟹ (v), we suppose that (viii) holds. Thus we assume that (3.73) holds with appropriate k, l′, {t_{i}}, {λ_{i}}, {t′_{j}} and {τ′_{j}} as stated in (viii). Without loss of generality, assume that {t′_{1}, ⋯, t′_{l}} ⊆ B^{rb}(p^{∗}), {t′_{l+1}, ⋯, t′_{l′}} ⊆ B(p^{∗}) \ B^{rb}(p^{∗}). Note that if l + 1 ≤ j ≤ l′, then t_{j} ∈ Q_{S} ∪ Q_{E}, and Ω_{t′_{j}} is either a singleton or a line-segment containing p^{∗}(t_{j}) as an internal point (seeing (3.9). Hence

Re \overline{τ′_{j}}α = 0 for each α ∈ span_{R}(Ω_{t′_{j}} − p^{∗}(t′_{j})), j = l + 1, ⋯, l′.  (3.74)

This implies that, for each p ∈ P_{R}, Re \overline{τ′_{j}}p(t′_{j}) = 0 if l + 1 ≤ j ≤ l′ because p(t′_{j}) ∈ span_{R}(Ω_{t′_{j}} − p^{∗}(t′_{j})) by (3.10). Consequently, (3.73) implies that

Re ∑_{i=1}^{k} λ_{i}p(t_{i})\overline{σ(t_{i})} + Re ∑_{j=1}^{l} p(t′_{j})\overline{τ′_{j}} = 0 for each p ∈ P_{R}.  (3.75)

Replacing λ_{i}, t′_{j} by their appropriate positive multipliers if necessary, we can assume that k+l ≤ m+1. To see this, we note first that if \frac{τ′_{j}}{|τ′_{j}|} ∈ τ(t′_{j}) \ τ_{r}(t′_{j}), then (3.16), (3.13) and (3.10) imply that Re p(t′_{j})\overline{τ′_{j}} = 0 for each p ∈ P_{R} and thus the corresponding term in (3.75) can be deleted. Henceforth we suppose therefore that each \frac{τ′_{j}}{|τ′_{j}|} ∈ τ_{r}(t_{j}) in (3.75). Noting that k ≥ 1 from the assumption and recalling the definition (3.20) and (3.25), it follows from (3.75) (applied to ψ_{1}, ⋯, ψ_{m} in place of p) that

−b_{r}(t_{1}) ∈ cone {b_{r}(t_{2}), ⋯, b_{r}(t_{k}), c_{r}(t′_{1}), ⋯, c_{r}(t′_{l})} ⊆ ℝ^{m}.

Consequently, by [19, Corollary 17.1.2], −b_{r}(t_{1}) can be expressed as a linear combination of at most m elements from {b_{r}(t_{2}), ⋯, b_{r}(t_{k}), c_{r}(t′_{1}), ⋯, c_{r}(t′_{l})} with positive coefficients. Thus, appropriately redefine λ_{i} and t′_{j} if necessary, we can assume that, k+l ≤ m+1, (3.75) holds for each p ∈ {ϕ_{1}, ⋯, ϕ_{m}} and hence for all p ∈ P_{R}. Therefore (viii) ⟹ (vii) and hence (viii) ⟺ (vii).

For (vii)⟹(v) & (i), suppose that (3.70) holds with appropriate {t_{i}}, {λ_{i}}, {t′_{j}} and {τ′_{j}} given in (vii). By an earlier argument, we may assume that {t′_{1}, ⋯, t′_{r}} ⊆ Q_{N}, {t′_{r+1}, ⋯, t′_{l}} ⊆ Q_{E} and \frac{τ′_{j}}{|τ′_{j}|} ∈ τ_{r}(t_{j}) for each r + 1 ≤ j ≤ l. Thus, (3.70) means that the origin of ℝ^{m} belongs to the convex hull of the W_{r}. Consequently, (v) holds. We go on to show that (i) holds. To this end, let p ∈ P_{Ω}. Then Then p^{∗} − p ∈ P_{R} and

Re ∑_{i=1}^{k} λ_{i}(p^{∗} − p)(t_{i})\overline{σ(t_{i})} + Re ∑_{j=1}^{l}(p^{∗} − p)(t′_{j})\overline{τ′_{j}} = 0.  (3.76)

Since k ≥ 1 and each λ_{i} > 0, we assume without loss of generality that ∑_{i=1}^{k} λ_{i} = 1. Thus ∥f − p∥ ≥ ∑_{i=1}^{k} λ_{i}|(f − p)(t_{i})|^{2}. Since p ∈ P_{Ω} and τ′_{j} ∈ −N_{Ω_{t′_{j}}} (p^{∗}(t′_{j})), one has that

Re (p^{∗} − p)(t′_{j})\overline{τ′_{j}} ≤ 0, j = 1, 2, ⋯, l.  (3.77)

Hence

∥f − p∥^{2} ≥ ∑_{i=1}^{k} λ_{i}|(f − p)(t_{i})|^{2} + 2Re ∑_{j=1}^{l}(p^{∗} − p)(t′_{j})\overline{τ′_{j}}
= ∑_{i=1}^{k} λ_{i}|(f − p^{∗})(t_{i})|^{2} + ∑_{i=1}^{k} λ_{i}|(p^{∗} − p)(t_{i})|^{2}

+2Re ∑_{i=1}^{k} λ_{i}(p^{∗} − p)(t_{i})\overline{(f − p^{∗})(t_{i})} + 2Re ∑_{j=1}^{l}(p^{∗} − p)(t′_{j})\overline{τ′_{j}}

= ∑_{i=1}^{k} λ_{i}|(f − p^{∗})(t_{i})|^{2} + ∑_{i=1}^{k} λ_{i}|(p^{∗} − p)(t_{i})|^{2}
≥ ∥f − p^{∗}∥^{2},

where the second equality hods because of (3.76) while the last inequality holds because {t_{i}} ⊆ M(σ). This means that p^{∗} is a best approximation to f from P_{Ω} and hence (i) holds.

For (v)⟹(vi) & (ii), suppose that there exist nonnegative integers k, l with k + l ≥ 1 such that

0 ∈ conv {b_{r}(t_{1}), b_{r}(t_{2}), ⋯, b_{r}(t_{k}), c_{r}(t′_{1}), ⋯, c_{r}(t′_{l})} ⊆ ℝ^{m}  (3.78)

for some {t_{i}}_{i=1}^{k} ⊆ M(σ) and {t′_{j}}_{j=1}^{l} ⊆ B^{rb}(p^{∗}). By the Caratheodory Theorem (cf. [2] or [21, p. 73]), we assume without loss of generality that k + l ≤ m + 1. Moreover by (3.17), (3.20) and (3.25), there exist {λ_{i}} ⊂ (0, +∞) and {τ′_{j}} ⊂ ℂ \ {0} with τ′_{j} ∈ −N_{Ω_{t′_{j}}} (p^{∗}(t′_{j})) \ {0} for each j such that (3.70) holds for each p ∈ {ψ_{1}, ⋯, ψ_{m}} and hence for each p ∈ P_{R}. (Caution: Since k may be zero, we cannot conclude (vii) to hold.) Now by applying Lemma 3.6 to the functional Φ: P_{R} → ℂ defined by

Φ(p) = ∑_{i=1}^{k} λ_{i}p(t_{i})\overline{σ(t_{i})} + ∑_{j=1}^{l} p(t′_{j})\overline{τ′_{j}} for each p ∈ P

we conclude that (3.57) holds with appropriate {t′′_{j}}, {τ′′_{j}} stated in Lemma 3.6. By the the Caratheodory Theorem, we assume that k + l + s ≤ 2n + 1. Thus we see (vi) holds (dividing both sides of (3.57) by a positive constant if necessary). Note, in passing, again that (viii) would hold provided that k ≠ 0. Moreover (ii) must also hold because otherwise there exists an element p_{0} ∈ P_{R} such that

max\{Re (p_{0}(t_{i})\overline{σ(t_{i})}), Re (p_{0}(t′_{j})\overline{τ′_{j}})\} < 0 for each i = 1, ⋯, k and j = 1, ⋯, l.  (3.79)

This contradicts (3.70) as the number of the left-hand side of (3.70) with p = p_{0} is negative by (3.79). Hence, the proof of (v)⟹(vi) & (ii) is complete.

Finally we show that (viii)⟺(ix). Suppose first that (ix) holds. Then, there exist v^{∗} ∈ J(σ), w_{j}^{∗} ∈ N_{C_{t′_{j}}} (p^{∗}) and λ′_{j} > 0, j = 1, 2, ⋯, s with p^{∗} ∈ bd C_{t′_{j}} (namely t′_{j} ∈ B(p^{∗}) by (3.30)) such that

⟨v^{∗}, p⟩ = ∑_{j=1}^{s}⟨w_{j}^{∗}, p⟩ for all p ∈ P.  (3.80)

Set u^{∗} = v^{∗}/∥v^{∗}∥. Applying [22, Lemma 1.3, p.169] to the real linear span of P ∪ {f}, there exist a positive integer r (with 1 ≤ r ≤ 2(n + 1)), r extreme points u_{1}^{∗}, ⋯, u_{r}^{∗} of the unit ball Σ^{∗} of C(Q)^{∗} and positive constants β_{i}, i = 1, 2, ⋯, r, with ∑_{i=1}^{r} β_{i} = 1 such that

⟨u^{∗}, p⟩ = ∑_{i=1}^{r} β_{i}⟨u_{i}^{∗}, p⟩ for all p ∈ span (P ∪ {f}).  (3.81)

By a well-known representation of the extreme points of Σ^{∗} (cf. [22, p.69]), there exist some α_{i} ∈ ℂ with |α_{i}| = 1 and t_{i} ∈ Q such that

u_{i}^{∗} = α_{i}e_{t_{i}}, i = 1, 2, ⋯, r.

By the definition of u^{∗}, ∥u^{∗}∥ = 1 and ⟨u^{∗}, σ⟩ = ∥σ∥; hence, by (3.81), t_{i} ∈ M(σ) and α_{i} = \overline{(σ)(t_{i})}/∥σ∥. Furthermore, by (3.41), for each j, there exists τ′_{j} ∈ −N_{Ω_{t′_{j}}} (p^{∗}(t′_{j})) such that −w_{j}^{∗} = \overline{τ′}_{j}e_{t′_{j}}. Therefore, (3.80) becomes

∑_{i=1}^{r} β′_{i}p(t_{i})\overline{σ(t_{i})} + ∑_{j=1}^{s} p(t′_{j})\overline{τ}′_{j} = 0 for all p ∈ P,  (3.82)

where β′_{i} = ∥v^{∗}∥β_{i}/∥σ∥ for each i = 1, ⋯, r. Set

c_{j} = (ϕ_{1}(t), ⋯, ϕ_{n}(t))\overline{τ}′_{j} for each j = 1, ⋯, s.

Then (3.82) implies that

−β′_{1}b(t_{1}) ∈ cone{β′_{2}b(t_{2}), ⋯, β′_{r}b(t_{r}), c_{1}, ⋯, c_{s}}.

Since dim_{R}P = 2n, by [19, Corollary 17.1.2], −β′_{1}b(t_{1}) can be expressed as a linear combination of at most 2n elements from {β′_{2}b(t_{2}), ⋯, β′_{r}b(t_{r}), c_{1}, ⋯, c_{s}} with positive coefficients. Hence, replacing β′_{i} and τ′_{j} by appropriate positive multiplies of them we can assume without loss of generality that r, s in (3.82) satisfy the additional property that 1+s ≤ r+s ≤ 2n+1. Thus (viii) holds with (k, l′) = (r, s). Conversely, suppose that (viii) holds. Hence we have (3.73) with appropriate {t_{i}}_{i=1}^{k}, {λ_{i}}_{i=1}^{k} and {t′_{j}}_{j=1}^{l′}, {τ′_{j}}_{j=1}^{l′} as stated in (viii). We can of course assume that ∑_{i=1}^{k} λ_{i} = 1, and rewrite (3.73) as

∑_{i=1}^{k} λ_{i}\overline{σ(t_{i})}e_{t_{i}} = − ∑_{j=1}^{l′} \overline{τ′_{j}}e_{t′_{j}} (∈ P^{∗}).  (3.83)

By Lemma 3.2, \overline{τ′_{j}}e_{t′_{j}} ∈ N_{C_{t′_{j}}} (p^{∗}) for each j = 1, 2, ⋯, l′. On the other hand, since t_{i} ∈ M(σ), we

have that ⟨\overline{σ(t_{i})}e_{t_{i}}, σ⟩ = ∥σ∥^{2} for each i = 1, 2, ⋯, k. Therefore the functional expressed by either side of (3.83) belongs to the intersection in (ix).

Theorem 3.2. It holds that (v)⟺(vii) if IC is assumed, and that (vi)⟺(viii) if IC_{0} is assumed.

Proof. Suppose that (v) holds and proceeds as in the proof for (v)⟹(vi) & (ii) of Theorem 3.1. If, IC is assumed in addition, 0 ∉ conv U_{r} by Lemma 3.1. Hence k in (3.78) must be non-zero and so (vii) holds. Similarly, suppose that (vi) holds (thus, with the exception that k is possible zero, (3.73) holds). Suppose further that IC_{0} is assumed (instead of IC). Then, 0 ∉ conv U by Lemma 3.1. Hence k in (3.73) must be non-zero. Therefore (viii) holds.

Theorem 3.3. If the system {P, C_{t} : t ∈ Q} has the strong CHIP at p^{∗}, then (i)⟺(vii).

Proof. Note that P_{Ω} = P ∩ (∩_{t∈Q}C_{t}). By the implication (i)⟹(iv) in Theorem 2.3 and the fact that P is a vector subspace containing p^{∗} (so N_{P}(p^{∗})|_{P} = 0), we now have that (i)⟺(ix) thanks to the strong CHIP assumption. Since (ix)⟺(vii) by Theorem 3.1, (i)⟺(vii) holds.

Theorem 3.4. If both LKC and IC are assumed, then the statements in the list (i)–(ix) but (vi) are equivalent to each other.

Proof. Suppose that both LKC and IC hold. we will show that the CCS-system {P, C_{t} : t ∈ Q} has the strong CHIP at p^{∗}. For this purpose, note that, by Lemma 3.3 and Remark 2.1, the condition LKC implies that the set-valued function t ↦ C_{t} is lower semicontinuous on Q while, by Lemma 3.1, the condition IC implies that the system {P, C_{t} : t ∈ Q} satisfies the weak-strong interior-point condition with (Q_{E}, Q_{S}). By Theorem 2.1, the system {P, C_{t} : t ∈ Q} has the strong CHIP at p^{∗}. By Theorems 3.3, 3.1 and 3.2, it remains to show that (ii)⟹(v). Suppose on the contrary that (ii) holds but (v) is false. Then, by Lemma 3.5, 0 ∉ conv \overline{W_{r}^{+}} (⊆ conv W_{r}). By the Linear Inequality Theorem (see [2]), there exists z^{0} = (γ_{1}^{0}, ⋯, γ_{m}^{0}) ∈ ℝ^{m} such that

⟨u, z^{0}⟩ < 0 for all u ∈ \overline{W_{r}^{+}}.  (3.84)

Then max_{u∈\overline{W_{r}^{+}}}⟨u, z^{0}⟩ < 0 because \overline{W_{r}^{+}} is compact (noting that W_{r}^{+} is bounded). Let p_{0} = ∑_{i=1}^{m} γ_{i}^{0}ψ_{i}. Then p_{0} ∈ P_{R}. By (3.25) and (3.21), for any t ∈ M(σ), t′ ∈ B^{rb}(p^{∗}) and τ ∈ τ_{r}^{+}(t′) and

Re (p_{0}(t)\overline{σ(t)}) = ⟨b_{r}(t), z^{0}⟩, Re (p_{0}(t′)\overline{τ}) = ⟨u_{τ} , z^{0}⟩,

where u_{τ} ∈ c_{r}^{+}(t′) is defined by u_{τ} := (Re ψ_{1}(t′)\overline{τ}, ⋯, Re ψ_{m}(t′)\overline{τ}). Since \{b_{r}(t)\} ∪ c_{r}^{+}(t′) ⊆ W_{r}^{+}, we have that

max\{Re (p_{0}(t)\overline{σ(t)}), max_{τ∈τ_{r}^{+}(t′)} Re (p_{0}(t′)\overline{τ})\} = max\{⟨b_{r}(t), z^{0}⟩, max_{τ∈τ_{r}^{+}(t′)}⟨u_{τ} , z_{0}⟩\} ≤ max_{u∈\overline{W_{r}^{+}}}⟨u, z^{0}⟩ < 0,

which contradicts (ii).

Theorem 3.5. If both KC and IC_{0} are assumed, then the statements (i)–(ix) are mutually equivalent.

Proof. Suppose that both KC and IC_{0} hold. Then W is compact in ℂ^{n} by Lemma 3.4. Using this, and similar arguments as in the proof of (ii)⟹(v)in Theorem 3.4 gives that (ii)⟹(vi) (use W, ℂ^{n} and Re ⟨u, z⟩ to replace \overline{W_{r}^{+}}, ℝ^{m} and ⟨u, z⟩). By Theorem 3.2, (vi)⟹(viii). Thus, by Theorem 3.1, it remains to show (i)⟺(vii). In view of Theorem 3.3, it suffices to show that the CCS-system {P, C_{t} : t ∈ Q} has the strong CHIP at p^{∗}. But this follows easily from Theorem 2.2 which is applicable to this system by Lemma 3.1 (i) and Lemma 3.3 (thanks to the assumptions).

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