Nonlinear Analysis 58 (2004) 367 – 377
www.elsevier.com/locate/na
Ambiguous loci of mutually nearest and mutually
furthest points in Banach spaces
Chong Lia , Hong-Kun Xub;∗
a Department
b School
of Mathematics, Zhejiang University, Hangzhou 310027, People’s Republic of China
of Mathematical Sciences, University of KwaZulu-Natal, Westville Campus, Private Bag
X54001, Durban 4000, South Africa
Received 7 August 2003; accepted 12 April 2004
Abstract
Let X be a real separable strictly convex Banach space and G a nonempty closed subset
of X . Let K(X ) (resp. Kb (X )) denote the family of all nonempty boundedly compact (resp.
compact) convex subsets of X endowed with the H -topology (resp. the Hausdor1 distance),
KG (X ) (resp. KGb (X )) the closure of the set {A ∈ K(X ) : A ∩ G = ∅} (resp. {A ∈ Kb (X ) :
A ∩ G = ∅}), and V(G) (resp. Vb (G)) the family of A ∈ KG (X ) (resp. A ∈ KGb (X )) such that
the minimization problem min(A; G) fails to be well-posed. It is proved that for most (in the
sense of the Baire category) closed subsets (resp. bounded closed subsets) G of X , V(G) (resp.
Vb (G)) is everywhere uncountable in KG (X ) (resp. KGb (X )). A similar result for the mutually
furthest point problem is also given.
? 2004 Elsevier Ltd. All rights reserved.
MSC: primary 41A65; 54E52; secondary 46B20
Keywords: Minimization problem; Maximization problem; Well-posed; Ambiguous loci
1. Introduction
Let X be a real Banach space. We shall use the following notations for various
families of subsets of X .
A(X )—the family of all nonempty closed subsets,
Ab (X )—the family of all nonempty closed bounded subsets,
Supported in part by the National Natural Science Foundations of China (Grant Nos. 10271025).
∗
Corresponding author. Fax: +27-31-204-4806.
E-mail addresses: cli@zju.edu.cn (C. Li), xuhk@ukzn.ac.za (H.-K. Xu).
0362-546X/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.na.2004.04.008
368
C. Li, H.-K. Xu / Nonlinear Analysis 58 (2004) 367 – 377
C(X )—the family of all nonempty closed convex subsets,
Cb (X )—the family of all nonempty closed bounded convex subsets,
K(X )—the family of all nonempty convex boundedly compact subsets,
Kb (X )—the family of all nonempty convex compact subsets.
Let G; A ∈ A(X ). DeCne
AG := inf {z − x : x ∈ A; z ∈ G}
and, if G and A are additionally bounded
AG := sup{z − x : x ∈ A; z ∈ G}:
Given a nonempty closed (resp. closed bounded) subset G of X , according to [8],
for A ∈ A(X ) (resp. A ∈ Ab (X ), a pair of (x0 ; z0 ) with x0 ∈ A; z0 ∈ G is called a solution of the minimization (resp. maximization) problem, denoted by min(A; G) (resp.
max(A; G)), if x0 −z0 =AG (resp. x0 −z0 =AG ). Moreover, any sequence {(x n ; zn )}
with x n ∈ A; zn ∈ G, such that limn→∞ x n −zn =AG (resp. limn→∞ x n −zn =AG ) is
called a minimizing (resp. maximizing) sequence for min(A; G) (resp. max(A; G)). A
minimization (resp. maximization) problem is said to be well-posed if it has a unique
solution and every minimizing (resp. maximizing) sequence converges strongly to the
solution.
Recall that the Hausdor1 distance on the space Ab (X ) is deCned by
H (A; B) = max sup inf a − b; sup inf a − b ; A; B ∈ Ab (X ):
a∈A b∈B
b∈B a∈A
It is known that (Ab (X ); H ) is a complete metric space.
De Blasi et al. [8] considered the well-posedness of the minimization and maximization problems. They proved that if X is a uniformly convex Banach space, then the set
of A ∈ CGb (X ) (resp. A ∈ Cb (X )) such that the minimization problem min(A; G) (resp.
maximization max(A; G)) is well-posed, is a dense G -subset of CGb (X ) (resp. Cb (X )),
where CGb (X ) stands for the closure of the set {A ∈ Cb (X ) : AG ¿ 0}. Recently, the
Crst author of the present paper extended the above result to the framework of reHexive
locally uniformly convex Banach spaces for the class K(X ) in [13]. These are positive
results on the well-posedness. For some motivating ideas and related works see also
[1,2,6,7,9–12,14,15].
In the present paper, inspirit of some works on the nearest point and furthest problems from [3–5,16], we will establish some negative results on the well-posedness.
More precisely, let V(G) (resp. M(G)) denote the set of all A ∈ KG (X ) (resp.
A ∈ Kb (X )) such that the minimization problem min(A; G) (resp. maximization problem max(A; G)) fails to be well-posed, that is,
V(G) = {A ∈ KG (X ) : min(A; G) is not well-posed};
(1.1)
where KG (X ) = {A ∈ K(X ) : AG ¿ 0} and
M(G) = {A ∈ Kb (X ) : max(A; G) is not well-posed}:
(1.2)
C. Li, H.-K. Xu / Nonlinear Analysis 58 (2004) 367 – 377
369
We shall prove that if X is a strictly convex and separable Banach space, then the
family of nonempty closed subsets (resp. bounded closed subsets) G ∈ A(X ) such
that V(G) (resp. M(G)) is everywhere uncountable in KG (X ) (resp. Kb (X )) is
residual in A(X ) (resp. Ab (X )), where A(X ) is endowed with the H -topology while
Ab (X ) with the Hausdor1 metric. Our results generalize some results on the nearest
and furthest point problems from [4,5,16]. Moreover, it should be remarked that all
known results on ambiguous loci in best approximation theory were established for
some families of bounded subsets of X . Our results mutually nearest point problems
obtained in the present paper are for some families of unbounded subsets of X .
We conclude this section with some notation. If X is a Banach space and if A ⊂ X ,
then AK stands for the closure of A, diam(A) for the diameter of A, and coA for the
closed convex hull of A. We use B(x; r) to denote the closed ball with centre x and
radius r in X , in particular, B stands for B(0; 1). We also need the following deCnition
of everywhere uncountability in a topological space, see [3–5] for the deCnition in a
metric space.
Denition 1.1. Let S be a subset of the topological space E. S is called everywhere
uncountable in E if, for every x ∈ E and its neighbourhood U (x), the set of the intersection, S ∩ U (x), is nonempty and uncountable.
Note that if A is everywhere uncountable, then it is dense in E; the converse
is however not true, in general.
2. Ambiguous loci of mutually nearest points
We begin with the following deCnition. Recall that the Hausdor1 metric can also
be redeCned by
H (A; B) = max{e(A; B); e(B; A)};
A; B ∈ Ab (X );
(2.1)
where e(A; B) is deCned by
e(A; B) = inf { ¿ 0 : B ⊆ A + B}:
(2.2)
Furthermore, for any ¿ 0, A; B ∈ A(X ), deCne
H (A; B) = max{e(A; B ∩ B); e(B; A ∩ B)}:
(2.3)
Then it is easy to see that, for any ¿ 0 and A; B ∈ A(X ),
H (A; B) 6 H (A; B)
(2.4)
if H (A; B) is allowed to be +∞.
Denition 2.1. The H -topology on A(X ) is deCned as follows: for any A ∈ A(X ),
the neighbourhood basis of A is the family of all subsets U (A; r) with ¿ 0 and r ¿ 0
deCned by
U (A; r) = {B ∈ A(X ) : H (A; B) ¡ r}:
(2.5)
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C. Li, H.-K. Xu / Nonlinear Analysis 58 (2004) 367 – 377
Throughout the whole paper, we always endow A(X ) and Ab (X ) with the H topology and Hausdor1 metric, respectively. Thus, K(X ) is a topological subspace of
A(X ) while Kb (X ) is a metric subspace of Ab (X ). For F; G ∈ A(X ), let
AF (X ) := {G ∈ A(X ) : FG ¿ 0};
(2.6)
where the closure is taken in the topological space A(X ). Recall that the set V(G)
is deCned by
V(G) = {A ∈ KG (X ) : min(A; G) is not well-posed};
where KG (X ) = K(X ) ∩ AG (X ) = {A ∈ K(X ) : AG ¿ 0}.
Theorem 2.1. Suppose that X is a separable strictly convex Banach space. Then
the set
A∗ (X ) := {G ∈ A(X ) : V(G) is everywhere uncountable in KG (X )}
is residual in A(X ).
Proof. For r ¿ 0, ¿ 0 and F ∈ K(X ), deCne
AF; r; := {G ∈ AF (X ) : V(G) ∩ U (F; r) is countable in KG (X )};
(2.7)
where U (F; r) is deCned by U (F; r) = {B ∈ A(X ) : H (F; B) ¡ r}. We will prove
that AF; r; is nowhere dense in A(X ). For the purpose, let G ∈ AF; r; . Since X is
strictly convex and F is boundedly compact, without loss of generality, we may assume
that min(F; G) has a unique solution (fF ; gF ). We may also assume FG ¿ 0. (If,
otherwise, FG = 0, we can take G̃ which is near G, such that min(F; G̃) has a unique
solution and F G̃ ¿ 0.) It is suMcient to show that, for any ¿ 0, ¿ 0, there exist
Y ∈ U (G; ) ∩ AF; r; and ¿ 0, ¿ 0 such that U (Y; ) ∩ AF; r; = ∅.
Without loss of generality, let ¿ 0 and let satisfy that 0 ¡ ¡ min{r; FG }. Let
(gF − fF ):
g1 = fF + 1 −
(2.8)
2FG
It is immediately clear that
d(g1 ; F) = fF − g1 = FG − =2 ¿ =2
(2.9)
and
d(g1 ; G) = gF − g1 = =2:
(2.10)
Let next g2 ∈ X be such that
g1 − g2 = =2
and
d(g2 ; F) = d(g1 ; F);
(2.11)
where d(gi ; F) denote the distance from gi to F. Note that g1 ; g2 ; gF are not colinear
and X is strictly convex. Then, by (2.10) and (2.11),
d(g2 ; G) 6 g2 − gF ¡ g2 − g1 + g1 − gF = :
Put
Y := G ∪ {g1 ; g2 }:
(2.12)
C. Li, H.-K. Xu / Nonlinear Analysis 58 (2004) 367 – 377
We now estimate by (2.10) and (2.12)
H (G; Y ) = max
371
sup d(g; Y ); sup d(y; G)
g∈G
y∈Y
= sup d(y; G)
y∈Y
= max{d(g1 ; G); d(g2 ; G)}
¡ :
This with (2.4) implies that Y ∈ U (G; ).
Let now f1 := fF and f2 ∈ F be the unique best approximation to g2 from F. Set
;
(2.13)
+ :=
8d(g1 ; F)
ui := (1 − )fi + gi ;
0 6 6 1;
i = 1; 2
(2.14)
and
h :=
1
min min{d(g1 ; F2 ) − d(g2 ; F2 ); d(g2 ; F1 ) − d(g1 ; F1 )};
2 ∈[+=2;+]
(2.15)
where
Fi := co(F ∪ {ui });
i = 1; 2:
(2.16)
Fi
Then
is boundedly compact. We now show that h ¿ 0. Assume the contrary that
h 6 0. With no loss of generality, we may assume that
d(g1 ; F2 ) 6 d(g2 ; F2 ):
(2.17)
K
K
Let fK be the unique point in F2 such that g1 − f=d(g
1 ; F2 ). Write f =(1− tK)f + tKu2
K
for some t ∈ [0; 1] and f ∈ F to get
K F) 6 fK − (1 − tK)f − tKf2 = tKu2 − f2 = tKf2 − g2 = tKd(g2 ; F)
d(f;
6 tKd(u2 ; F):
(2.18)
It turns out, by (2.17) and (2.11), that
K + d(f;
K F)
d(g1 ; F) 6 g1 − f
K F)
= d(g1 ; F2 ) + d(f;
K F)
6 d(g2 ; F2 ) + d(f;
6 g2 − u2 + tKd(u2 ; F)
6 g2 − u2 + d(u2 ; F)
= d(g2 ; F) = d(g1 ; F):
So we must have tK = 1 and fK = u2 , resulting by (2.17) again in that
g1 − u2 = d(g1 ; F2 ) 6 g2 − u2 :
(2.19)
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C. Li, H.-K. Xu / Nonlinear Analysis 58 (2004) 367 – 377
Noting by (2.11)
g1 − f2 ¿ d(g1 ; F) = d(g2 ; F) = g2 − f2 (2.20)
we obtain by (2.19) that
g1 − f2 = g1 − u2 + u2 − f2 6 g1 − u2 + u2 − f2 6 g2 − u2 + u2 − f2 = g2 − f2 6 g1 − f2 :
Hence we have
g1 − u2 + u2 − f2 = g1 − u2 + u2 − f2 (2.21)
and, since X is strictly convex
g1 − u2 = (u2 − f2 )
(2.22)
for some ¿ 0. This contradicts the fact that g1 ; f2 ; u2 are not colinear.
Now let
;h
and = 2 max f + max (fi + gi ) + 1 :
:= min
i=1;2
f∈F
8
(2.23)
For any Z ∈ U (Y; ), we get
d(gi ; Z) 6 H (Y; Z) ¡ ;
i = 1; 2
(2.24)
as gi ∈ Y ∩ B. Set
Zi := B(gi ; ) ∩ Z;
i = 1; 2:
(2.25)
Then, by (2.24) and (2.25), Zi = ∅ for i = 1; 2 and by (2.11) and (2.23), Z1 ∩ Z2 = ∅.
Let
At := co(F ∪ {(1 − t)u1 + tu2 });
Then, for each 0 6 t 6 1; +=2 6 6 +,
At Z = At (Z1 ∪Z2 ) ;
0 6 t 6 1; +=2 6 6 +:
At
(2.26)
is boundedly compact. We next show that
0 6 t 6 1 and +=2 6 6 +:
(2.27)
Indeed, for each a := (1 − s)f + s[(1 − t)u1 + tu2 ] ∈ At , where 0 6 s 6 1, 0 6 t 6 1,
+=2 6 6 +, f ∈ F, we have
d(a; F) 6 a − (1 − s)f − s[(1 − t)f1 + tf2 ] 6 max{f1 − u1 ; f2 − u2 }
6 d(g1 ; F):
(2.28)
Similarly, for i = 1; 2 and each b := (1 − s)f + sui ∈ Fi ,
d(b; F) 6 b − [(1 − s)f + sfi ] = sfi − ui = sd(ui ; F) 6 d(gi ; F):
(2.29)
C. Li, H.-K. Xu / Nonlinear Analysis 58 (2004) 367 – 377
373
It follows from the inclusions F ⊆ At , F ⊆ Fi and (2.27), (2.28) that, for i = 1; 2
sup d(a; Fi ); sup d(b; At )
H (At ; Fi ) = max
b∈Fi
a∈At
sup d(a; F); sup d(b; F)
6 max
a∈At
b∈Fi
6 · max{d(g1 ; F); d(g2 ; F)}
= d(g1 ; F):
(2.30)
For i = 1; 2, by the deCnition of Fi , we obtain that
d(gi ; Fi ) = gi − ui = (1 − )d(gi ; F):
(2.31)
Hence, from (2.24), (2.30) and (2.31), we have that, for i = 1; 2
At Zi 6 d(gi ; At ) + d(gi ; Zi )
6 H (At ; Fi ) + d(gi ; Fi ) + 6 d(g1 ; F) + (1 − )d(gi ; F) + = FG − =2 + ;
(2.32)
where the last equality is because of (2.9) and (2.11).
Now, let z ∈ Z. Suppose that, for some i = 1; 2
d(z; At ) 6 At Zi :
(2.33)
Observe that
max a 6 max max a; max (fi + gi )
(2.34)
At Zi 6 FZi 6 max a − gi + d(gi ; Zi ) 6 max a + gi + 1:
(2.35)
a∈F
a∈At
i=1;2
and
a∈F
a∈F
It follows from (2.33) to (2.35) and (2.23) that
z 6 d(z; At ) + max a
a∈At
6 max a + gi + 1 + max max a; max (fi + gi )
a∈F
a∈F
i=1;2
6 2 max a + max (fi + gi ) + 1
a∈F
i=1;2
¡ :
Furthermore, since 6 +, by (2.4), (2.28) and (2.13),
H (At ; F) 6 H (At ; F) 6 sup{d(a; F) : a ∈ At } 6 d(g1 ; F) ¡ =8:
(2.36)
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C. Li, H.-K. Xu / Nonlinear Analysis 58 (2004) 367 – 377
Consequently,
GAt ¿ FG − H (At ; F) ¿ FG − =8:
(2.37)
Since H (Y; Z) ¡ and z ¡ , there exists y ∈ Y such that z − y ¡ so that
y = gi and y ∈ G. Thus, if z − gi ¿ , we have, by (2.37) and (2.32)
d(z; At ) ¿ d(y; At ) − ¿ GAt − ¿ GF − =8 − ¿ GF − =2 + ¿ At Zi
which contracts (2.33). This implies that z ∈ Zi , and hence (2.27) holds.
On the other hand, we have by deCnition of F1 Z1 6 d(g1 ; F1 ) + 6 d(g1 ; F1 ) +
1
[d(g2 ; F1 ) − d(g1 ; F1 )]
2
= d(g2 ; F1 ) +
1
[d(g1 ; F1 ) − d(g2 ; F1 )]
2
6 F1 Z2 + +
1
[d(g1 ; F1 ) − d(g2 ; F1 )]
2
6 F1 Z2 :
(2.38)
Similarly we have
F2 Z2 6 F2 Z1 :
(2.39)
For arbitrary Cxed ∈ [+=2; +], let
r(t) := Z1 At − Z2 At ;
0 6 t 6 1:
Then, by (2.38) and (2.39)
r(0) = F1 Z1 − F1 Z2 6 0;
r(1) = F2 Z1 − F2 Z2 ¿ 0:
It follows from the continuity of r(·) that there exists 0 6 t 6 1 such that r(t ) = 0;
hence
Z1 At = Z2 At
(2.40)
for each ∈ [+=2; +]. Since Z1 and Z2 are closed and disjoint, it follows that the problem
min(At ; Z) is not well-posed for each ∈ [+=2; +]. This means that At ∈ V(Z) for each
∈ [+=2; +]. In addition, by (2.36) and (2.37), At ∈ U (F; r) ∩ KG (X ). This implies that
V(Z) ∩ U (F; r) contains at least uncountable elements and consequently, Z ∈ AF; r; .
Therefore, AF; r; is nowhere dense in A as Z ∈ U (Y; ) is arbitrary.
Since the separability of X implies that K(X ) is separable, there exists a countable
subset S of K(X ) which is dense in K(X ). Let Q+ be the set of positive rationals.
DeCne
A˜ :=
AF; r; :
F∈S r;∈Q+
C. Li, H.-K. Xu / Nonlinear Analysis 58 (2004) 367 – 377
375
Then A˜ is of the Crst Baire category in A(X ). To complete the proof it remains to
prove that A∗ (X ) ⊇ A(X )\A.˜ To this end, let G ∈ A(X )\A.˜ For any r ¿ 0, ¿ 0 and
any F ∈ KG (X ), there exist r;
ˆ ˆ ∈ Q+ and F̂ ∈ S such that F̂ ∈ KG (X ) and Uˆ(F̂; r)
ˆ ⊆
U (F; r). Note that G ∈ A(X )\AF;ˆ r;ˆ ˆ. Since F̂ ∩ G = ∅, it follows that V(G) ∩ Uˆ(F̂; r)
ˆ
is nonempty and uncountable and so is V(G) ∩ U (F; r). This shows that V(G) is
everywhere uncountable in KG (X ) and the proof is complete.
Similarly, modifying slightly the proof of Theorem 2.1, we also have following
theorems. Let KGb (X ) denote the intersection set of Kb (X ) and CG (X ).
Theorem 2.2. Suppose that X is a separable strictly convex Banach space. Then the
set
Ab∗ (X ) := {G ∈ Ab (X ) : V(G)
is everywhere uncountable in KGb (X )}
is residual in Ab (X ).
Theorem 2.3. Suppose that X is a separable strictly convex Banach space. Then the
set
Aˆ∗ (X ) := {G ∈ Ab (X ) : V(G) is everywhere uncountable in KG (X )}
is residual in Ab (X ).
3. Ambiguous loci of mutually furthest points
We study in this section the ambiguous loci of mutually furthest points. For x ∈ X
and F ∈ Ab (X ), write e(x; F) = supf∈F x − f. Recall that the set M(G) is deCned
by
M(G) = {A ∈ Kb (X ) : max(A; G) is not well-posed}:
Theorem 3.1. Assume that X is a separable strictly convex Banach space. Then the
set
A∗b (X ) := {G ∈ Ab (X ) : M(G) is everywhere uncountable in Kb (X )}
is residual in Ab (X ).
Proof. Since the proof is similar to that of Theorem 2.1, we only sketch it here. For
F ∈ Kb (X ) and r ¿ 0, deCne
AF; r := {G ∈ Ab (X ) : M(G) ∩ U (F; r) is countable in Kb (X )};
where U (F; r)={B ∈ Ab (X ) : H (F; B) ¡ r}. We will show that AF; r is nowhere dense
in Ab (X ). In fact, let G ∈ AF; r . We may assume FG ¿ 0 and max(F; G) has a unique
solution (fF ; gF ). Let 0 ¡ ¡ min{r; FG } and
(gF − fF ):
g1 := fF + 1 +
2FG
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C. Li, H.-K. Xu / Nonlinear Analysis 58 (2004) 367 – 377
It is clear that e(g1 ; F) = fF − g1 = FG + =2. Let g2 ∈ X be such that
and e(g2 ; F) = e(g1 ; F)
g1 − g2 =
2
and put
Y := G ∪ {g1 ; g2 }:
Then we have that H (G; Y ) 6 since gi − gF 6 for i = 1; 2. Let f1 := fF and
f2 ∈ F be such that f2 − g2 = e(g2 ; F). Set
+ :=
;
8e(g1 ; F)
ui := (1 + )fi − gi ;
0 6 6 1;
i = 1; 2
and
h :=
1
min min{e(g2 ; F2 ) − e(g1 ; F2 ); e(g1 ; F1 ) − e(g2 ; F1 )};
2 ∈[+=2;+]
where
Fi = co(F ∪ {ui });
i = 1; 2:
Then h ¿ 0. Now let
;h
:= min
8
and for Z ∈ U (Y; ), set
Zi := B(gi ; ) ∩ Z;
i = 1; 2:
Then Zi = ∅ for i = 1; 2 and Z1 ∩ Z2 = ∅. Let
At := co(F ∪ {(1 − t)u1 + tu2 });
0 6 t 6 1; +=2 6 6 +:
Then we have that for each ∈ [+=2; +] there exists 0 6 t 6 1 such that the problem
max(At ; Z) is not well-posed:
(3.1)
b
This implies that Z ∈ AF; r and AF; r is nowhere dense in A (X ) since Z ∈ U (Y; ) is
arbitrary.
Now let S be a countable dense subset for Kb (X ) and let Q+ be the set of positive
rationals. DeCne
A˜ :=
AF; r :
(3.2)
F∈S r∈Q+
Then A˜ is of the Crst Baire category in Ab (X ) and A∗ (X ) ⊇ Ab (X )\A.˜ Hence A∗b (X )
is residual in Ab (X ) and the proof is complete.
References
[1] G. Beer, R. Lucchetti, Convex optimization and the epi-distance topology, Trans. Am. Math. Soc. 327
(1991) 795–813.
C. Li, H.-K. Xu / Nonlinear Analysis 58 (2004) 367 – 377
377
[2] J.M. Borwein, S. Fitzpatrick, Existence of nearest points in Banach spaces, Can. J. Math. 41 (1989)
702–720.
[3] F.S. De Blasi, P.S. Kenderov, J. Myjak, Ambiguous loci of the metric projection onto compact starshaped
sets in a Banach space, Mh. Math. 119 (1995) 23–36.
[4] F.S. De Blasi, J. Myjak, Ambiguous loci in best approximation theory, in: S.P. Singh (Ed.),
Approximation Theory, Spline Functions and Applications, Kluwer Academic Publishers, Dordrecht,
1992, pp. 341–349.
[5] F.S. De Blasi, J. Myjak, Ambiguous loci of the nearest point mapping in Banach spaces, Arch. Math.
61 (1993) 377–384.
[6] F.S. De Blasi, J. Myjak, On a generalized best approximation problem, J. Approx. Theory 94 (1998)
54–72.
[7] F.S. De Blasi, J. Myjak, P.L. Papini, Starshaped sets and best approximation, Arch. Math. 56 (1991)
41–48.
[8] F.S. De Blasi, J. Myjak, P.L. Papini, On mutually nearest and mutually furthest points of sets in Banach
spaces, J. Approx. Theory 70 (1992) 142–155.
[9] A. Dontchev, T. Zolezzi, Well Posed Optimization Problems, in: Lecture Notes in Mathematics, Vol.
1543, Springer, Berlin, 1993.
[10] P.G. Georgiev, The strong Ekeland variational principle, the strong drop theorem and applications,
J. Math. Anal. Appl. 131 (1988) 1–21.
[11] S.V. Konjagin, On approximation properties of closed sets in Banach spaces and characterization of
strongly convex spaces, Sov. Math. Dokl. 21 (1980) 418–422.
[12] K.S. Lau, Almost Chebyshev subsets in reHexive Banach spaces, Indiana Univ. Math. J. 27 (1978)
791–795.
[13] C. Li, On mutually nearest and mutually furthest points in reHexive Banach spaces, J. Approx. Theory
103 (2000) 1–17.
[14] C. Li, On well posedness of best simultaneous approximation problems in Banach spaces, Sci. China
44 (2001) 1558–1570.
[15] C. Li, X.H. Wang, Almost Chebyshev set with respect to bounded subsets, Sci. China 40 (1997)
375–383.
[16] N.V. Zhivkov, Compacta with dense ambiguous loci of metric projections and antiprojections, Proc.
Am. Math. Soc. 123 (1995) 3403–3411.