Acta Mathematica Sinica, English Series
May, 2006, Vol. 22, No. 3, pp. 741–750
Published online: Nov. 10, 2005
DOI: 10.1007/s10114-005-0595-4
Http://www.ActaMath.com
On Generic Well-posedness of Restricted Chebyshev
Center Problems in Banach Spaces
Chong LI
Department of Mathematics, Zhejiang University, Hangzhou 310027, P. R. China
E-mail: cli@zju.edu.cn
Genaro LOPEZ
Department of Mathematic Analysis, University of Sevilla, Seville, Spain
E-mail: glopez@us.es
Abstract Let B (resp. K , BC , K C ) denote the set of all nonempty bounded (resp. compact,
bounded convex, compact convex) closed subsets of the Banach space X, endowed with the Hausdorff
metric, and let G be a nonempty relatively weakly compact closed subset of X. Let B o stand for the
set of all F ∈ B such that the problem (F, G) is well-posed. We proved that, if X is strictly convex and
Kadec, the set K C ∩ B o is a dense Gδ -subset of K C \ G. Furthermore, if X is a uniformly convex
Banach space, we will prove more, namely that the set B \ B o (resp. K \ B o , BC \ B o , K C \ B o )
is σ-porous in B (resp. K , BC , K C ). Moreover, we prove that for most (in the sense of the Baire
category) closed bounded subsets G of X, the set K \ B o is dense and uncountable in K .
Keywords Chebyshev center, Well-posedness, σ-porous, Ambiguous loci
MR(2000) Subject Classification 41A65, 41A50
1 Introduction
Let X be a real Banach space and let B (resp. K , BC , K C ) denote the set of all nonempty
bounded (resp. compact, bounded convex, compact convex) closed subsets of the Banach space
X, endowed with the Hausdorff metric, H(·, ·). As is well known, under such a metric, B is a
complete metric space. Obviously, X can be embedded as a subset of K in a natural way that,
for any x ∈ X, Ax ∈ K is defined by Ax = {x}.
For a closed subset G of X and an element x in X, an element g0 ∈ G is called a best
approximation to x from G if x − g0 ≤ x − g, ∀ g ∈ G. The set of best approximations
to x from G is denoted by PG (x), that is, PG (x) = {g0 ∈ G : x − g0 = dG (x)}, where
dG (x) = inf{x − g : g ∈ G} is the distance from x to G.
Following Dontchev and Zolezzi [1], the best approximation problem, denoted by min(x, G),
is well-posed, if x has a unique best approximation from G and each minimizing sequence of
x in G converges to the unique best approximation element. Thus, the problem min(x, G) is
called ill-posed if it is not well-posed. It is well known that, see for example [2], if X is a
strictly convex and Kadec–Banach space, and if G is a boundedly relatively weakly compact
closed subset of X, then the set of all x ∈ X such that the problem min(x, G) is well-posed is
a dense Gδ subset of X. Furthermore, the authors in [3] proved that the set of all x ∈ X such
that the problem min(x, G) is ill-posed is a σ-porous set in X provided that X is uniformly
convex, while the authors in [4, 5] established some generic results on ambiguous loci of the
Received August 21, 2003, Accepted June 18, 2004
The first author is supported in part by the National Natural Science Foundation of China (Grant No. 10271025).
The second author is supported in part by Projects BFM 2000-0344 and FQM-127 of Spain.
Li C. and Lopez G.
742
nearest point mapping. Then a natural question is of interest: Can these results be extended
to the case when x is replaced by a bounded set? One kind of such extensions is due to De
Blasi, Myjak and Papini in [6] where they studied a similar problem for the mutually nearest
point defined as follows. Let F ∈ BC and let λF G = inf{f − g : f ∈ F, g ∈ G}. Then
the minimization problem min(F, G) is called well-posed if it has a unique solution and every
minimizing sequence converges to the solution. Here (f0 , g0 ) with f0 ∈ F, g0 ∈ G is called a
solution of the problem min(F, G) if f0 −g0 = λF G , while a minimizing sequence (fn , gn ) with
fn ∈ F, gn ∈ G means that lim fn − gn = λF G . De Blasi, Myjak and Papini proved that if X
is a uniformly convex Banach space, then the set of all F ∈ BC G , such that the minimization
problem min(F, G) is well-posed, is a dense Gδ -subset of BC G , where BC G stands for the
closure of the set {A ∈ BC : λAG > 0}. Li [7] extended the above result to the framework of
reflexive locally uniformly convex Banach spaces for the class K C .
In the present paper, we will consider another extension; in other words, we will extend the
above results to the setting of restricted Chebyshev centers in Banach spaces. The setting is
as follows. For F ∈ B, an element g0 ∈ G is called a restricted Chebyshev center of F with
respect to G (or a best simultaneous approximation to F from G) if g0 is a solution of the
minimization problem, denoted by (F, G), defined as follows:
inf φF (g),
g∈G
where φF (g) := supf ∈F f − g. Let rG (F ) denote the restricted Chebyshev radius of F with
respect to G defined by rG (F ) = inf g∈G supf ∈F f −g. Then g0 is called a restricted Chebyshev
center of F with respect to G if and only if it satisfies that φF (g0 ) = rG (F ). Moreover, a
sequence {gn } ⊂ G satisfying limn→∞ supf ∈F f − gn = rG (F ) is called a minimizing sequence
of F in G. Thus the problem (F, G) is called well-posed if F has a unique restricted Chebyshev
center, denoted by gF , with respect to G and every minimizing sequence converges to gF .
The study of Chebyshev center problems dates back to at least 1962 (see [8]) and recently
it has been developed by many authors, see for example [9–12], etc. The problems considered
here are also in the spirit of Steckin [13], and some further researches in this direction can be
found in [14–20] and in the monograph [1]. The results of the present paper generalize some
results from [5, 11, 21, 22]
Now let us describe simply the main results of the paper. Let G be a fixed relatively weakly
compact, nonempty closed subset of a strictly convex and Kadec Banach space X, and let B o
stand for the set of all F ∈ B such that the problem (F, G) is well-posed. It is proved in the
next section that the set K C ∩ B o is a dense Gδ -subset of K C \ G. Furthermore, if X is a
uniformly convex Banach space, we will prove in Section 3 more, namely that the set B \ B o
(resp. K \ B o , BC \ B o , K C \ B o ) is σ-porous in B (resp. K , BC , K C ). Moreover, in
the last section, we also show that for most (in the sense of the Baire category) closed bounded
subsets G of X, the set K \ B o is dense and uncountable in K .
2 Almost Well-posedness
We begin with the refinements of some results on best approximation problems due to Borwein and Fitzpatrick [2]. We always assume that G is a closed nonempty subset and X0 is a
closed convex subset of X. We will use the following definition and proposition on the Frechet
differentiability of Lipschitz functions on an open set, see [23] and [24].
Definition 2.1 Let D be an open subset of X. A real-valued function f on D is said to be
Frechet differentiable at x ∈ D if there exists an x∗ ∈ X ∗ such that
f (y) − f (x) − x∗ , y − x
= 0,
lim
y→x
y − x
where x∗ is called the Frechet differential at x which is denoted by Df (x).
Proposition 2.1 Let f be a locally Lipschitz function on an open subset D of a Banach
Well-posedness of Chebyshev Center Problems
743
space with equivalent Frechet differentiable norm (in particular, X reflexive will do). Then f is
Frechet differentiable on a dense subset of D.
We still need some additional definitions.
Definition 2.2 [25] X0 is said to be strictly convex with respect to G (or G-strictly convex)
if for any x, y ∈ X0 with x − y ∈ G − G, x = y = 12 (x + y) implies that x = y. In
particular, we say X0 is strictly convex in the case when G = X.
Definition 2.3 X0 is said to be Kadec with respect to G (or G-Kadec) if for any sequence
{xn } ⊂ X0 , x ∈ X0 , xn → x weakly and xn → x, then xn −x → 0 provided xn −x ∈ G−G.
In particular, we say X0 is Kadec in the case when G = X.
Clearly, we have the following facts :
i) X is strictly convex ⇒ X0 is strictly convex ⇒ X0 is strictly convex with respect to G;
ii) X is Kadec ⇒ X0 is Kadec ⇒ X0 is Kadec with respect to G.
In the remainder of the paper, let U(x, r) (resp. B(x, r)) stand for the open (resp. closed)
ball around x of radius
⎧ r. Also let
⎫
∗
∗
∗
⎪
⎪
there
is
x
∈
X
with
x
=
1,
such
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎬
that for ∀ > 0, there is δ > 0 so that
.
Ω(G) = x ∈ X\G :
⎪
inf{x∗ , x − z : z ∈ G ∩ U(x, dG (x) + δ)} ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭
> (1 − )dG (x)
Then from the proofs of Theorem 6.1 and Corollary 5.8 in [2], we immediately have the
following Lemmas 2.1 and 2.2.
Lemma 2.1
Ω(G) ∩ X0 .
If X0 is G-strictly convex, then PG (x) is at most a singleton for any x ∈
Lemma 2.2 Let G be a closed, boundedly relatively weakly compact, nonempty subset of X.
If X0 is G-Kadec, then G is X0 ∩ Ω(G)-approximatively compact in the sense that for any
x ∈ X0 ∩ Ω(G), any minimizing sequence of x in G contains a subsequence which converges to
an element in G.
The following result describes the density of the set X0 ∩ Ω(G).
Lemma 2.3 Let G be a closed, boundedly relatively weakly compact, nonempty subset of X.
Then X0 ∩ Ω(G) is a dense Gδ -subset of X0 \G.
Proof From Theorem 5.12 in [2], it follows that Ω(G) is a Gδ subset of X0 \G. Thus it suffices to
prove that Ω(G) is a dense subset of X0 \G. To this end, let x0 ∈ X0 \G be arbitrary and suppose
dG (x0 ) > > 0. Fix N > x0 + dG (x0 ) + 2 and let K = weak − cl ({B(0, N ) ∩ G} ∪ {x0 }) .
Then K is weakly compact and if Y is the closed span of K, from the factorization theorem
of Davis, Figiel, Johnson and Pelczynski [26], there exist a reflexive Banach space R and a
one-to-one continuous linear mapping T : R → Y such that T (B(0, 1)) ⊇ K. Set OT = {y =
g + tx0 ∈ Y : g ∈ spanG, 0 < t < 1}. Then OT is open in Y so that O = {u ∈ R : T u ∈ OT }
is open in R. Define fK : O → [0, +∞) by fK (u) = dG (T u), ∀ u ∈ O. By Proposition 2.1, the
Lipschitz function fK is Frechet differentiable on a dense subset of O. The remainder of the
proof is the same as that of Lemma 5.15 in [2].
Combining Lemmas 2.1, 2.3 and 2.4, we obtain the following generic result on well-posedness
of the best approximation problems.
Proposition 2.2 Let G be a boundedly relatively weakly compact and nonempty closed subset
of X. If X0 is G-strictly convex and G-Kadec, then the set of all x ∈ X0 such that the problem
(x, G) is well-posed is a dense Gδ -subset of X0 \ G.
Now let us introduce the addition and multiplication in B. For any A, B ∈ B, λ > 0, define
A + B = {a + b : a ∈ A, b ∈ B},
λA = {λa : a ∈ A}.
Li C. and Lopez G.
744
Then, from [27], we have the embedding result of the space of bounded sets in X.
Lemma 2.4 There exists a Banach space (E, · ) such that K C is embedded as a convex
cone in E in such a way that :
(i) The embedding is isometric, that is, for any A, B ∈ K C , the Hausdorff metric H(A, B)
= A − B;
(ii) Addition and multiplication by nonnegative scalars in E induce the corresponding operations in K C ;
(iii) X is a linear subspace in E.
Furthermore, if X is reflexive, then the above statement is also true for BC .
In order to apply Proposition 2.2 to the restricted Chebyshev center problems, we must
verify the following two lemmas about the strict convexity and Kadec property of the embedding
of the sets K C and BC .
Lemma 2.5 Let G be a closed subset of X and let E be given by Lemma 2.4. If X is both
strictly convex and Kadec, then K C is G-strictly convex and G-Kadec in E.
Proof Let A, B ∈ K C with A − B = g − g0 for some g, g0 ∈ G and A = B = 12 A + B.
Then A = B + g − g0 and A + B = 2B + g − g0 . Take b0 ∈ B such that
2b0 + g − g0 = max 2b + g − g0 = 2B + g − g0 .
b∈B
Let a0 = b0 + g − g0 . Then a0 ∈ A and a0 + b0 = 2b0 + g − g0 = A + B, a0 ≤ A,
b0 ≤ B. It follows from the strict convexity of X that a0 − b0 = g0 − g = 0, which implies
that A = B. So K C is G-strictly convex.
Now let us prove K C is G-Kadec. Suppose that An ∈ K C , An = A0 , An −A0 ∈ G−G
for n = 0, 1, 2, . . . and An converges to A0 weakly in E. Then, taking x∗ ∈ E ∗ with x∗ =
1, x∗ (A0 ) = A0 , we have An + A0 ≥ x∗ (An + A0 ) → 2A0 , so that An + A0 → 2A0 . In
addition, there is hn ∈ G − G with An = A0 + hn for n = 0, 1, 2, . . .. Clearly, hn converges to 0
weakly in X. Hence 2A0 +hn → 2A0 . Now take an ∈ A0 such that 2an +hn = 2A0 +hn for n = 1, 2, . . .. Due to the compactness of A0 , we may assume, without loss of generality, that
an → a0 for some a0 ∈ A0 . Since
|2an + hn − 2a0 + hn | ≤ 2an − a0 → 0,
we have that
lim 2a0 + hn = lim 2an + hn = lim 2A0 + hn = 2A0 .
n
n
n
Note that 2a0 + hn ≤ a0 + a0 + hn ≤ 2A0 . It follows that a0 = A0 , a0 + hn →
A0 . Using the fact that X is Kadec, we have that hn → 0. This implies An − A0 → 0
and completes the proof.
Lemma 2.6 Let G be a closed nonempty subset of X and let E be given by Lemma 2.4. If
X is uniformly convex, then BC is G-strictly convex and G-Kadec in E.
Proof The proof of the strict convexity is similar to that in Lemma 2.5, and so we only
need to prove the G-Kadec property. For n = 0, 1, 2, . . ., let An , hn , an (n = 0) have the
same meaning as in the proof of Lemma 2.4. Then, an → A0 , an + hn → A0 and
an + (an + hn ) → 2A0 . It follows from the uniform convexity of X that hn → 0, which
yields An − A0 → 0 and the proof is complete.
Now we are ready to give the main theorems of this section, which are just the consequences
of Lemmas 2.4, 2.5 and Proposition 2.2.
Theorem 2.1 Let G be a boundedly relatively weakly compact and nonempty closed subset of
X. Suppose that X is strictly convex and Kadec. Then the set K C ∩ B o is a dense Gδ -subset
of K C \ G.
Theorem 2.2 Suppose X is uniformly convex. Let G be a closed nonempty subset of X.
Then the set BC ∩ B o is a dense Gδ -subset of BC \ G.
Well-posedness of Chebyshev Center Problems
745
3 Porosity of Ill-posedness
In this section, we will show the porosity of ill-posed sets. We begin with the notion of the
porosity, see for example [3]. Let (E, d) be a metric space and recall that Bd (x, r) denotes a
closed ball in E with radius r and center x.
Definition 3.1 A subset Y of E is said to be porous of E if there are 0 < t ≤ 1 and
r0 > 0 such that for every x ∈ E and r ∈ (0, r0 ], there is a point y ∈ E such that Bd (y, tr) ⊂
Bd (x, r) ∩ (E \ Y ). A subset Y is said to be σ-porous in E if it is a countable union of sets
which are porous in E.
Note that in this definition the statement “for every x ∈ E” can be replaced by “for every
x ∈ Y ”. Clearly, a set which is σ-porous in E is also meager in E, but the converse is false, in
general.
For F ∈ B, x ∈ X, r > 0, 1 > α > 0 and δ > 0, define
Mδ (F, x, r, α) = {y ∈ X : H(Fα , y) ≤ (1 − α)r + αrδ, H(F, y) ≥ r},
where Fα = (1 − α)F + αx.
Lemma 3.1 Suppose that X is a uniformly convex Banach space. Let 0 < r0 < +∞. Then
for any > 0 there exists δ0 > 0 such that for any 0 < δ < δ0 , diamMδ (F, x, r, α) < holds for
any F ∈ B, x ∈ X, r and α satisfying r ≤ H(F, x) ≤ (1 + δ)r, 0 < α < 1/2 and 0 < r ≤ r0 .
Proof Suppose on the contrary that for some > 0 and ∀ δ > 0, there exist F δ ∈ B, xδ ∈ X, rδ
and αδ satisfying rδ ≤ H(F δ , xδ ) ≤ (1 + δ)rδ , 0 < αδ < 1/2 and 0 < rδ ≤ r0 such that
diamMδ (F δ , xδ , rδ , αδ ) > 2. Then there exists yδ ∈ Mδ (F δ , xδ , rδ , αδ ) such that xδ − yδ > .
Clearly,
H(Fαδδ , xδ ) ≤ (1 − αδ )rδ (1 + δ), and H(Fαδδ , yδ ) ≤ (1 − αδ )rδ + αδ rδ δ,
where Fαδδ = (1 − αδ )F δ + αδ xδ .
Take fδ ∈ Fαδδ such that
fδ − αδ xδ
δ
1 − αδ − yδ ≥ H(F , yδ ) − αδ rδ δ ≥ rδ (1 − αδ δ).
That is,
αδ (fδ − xδ ) + (1 − αδ )(fδ − yδ ) ≥ (1 − αδ )rδ (1 − αδ δ).
Let x∗δ ∈ X ∗ , x∗δ = 1 such that
x∗δ , αδ (fδ − xδ ) + (1 − αδ )(fδ − yδ ) ≥ (1 − αδ )rδ (1 − αδ δ).
Then
x∗δ , αδ (fδ − xδ ) ≥ (1 − αδ )rδ (1 − αδ δ) − (1 − αδ )fδ − yδ ≥ (1 − αδ )rδ (1 − αδ δ) − (1 − αδ )rδ (1 − αδ + αδ δ)
= (1 − αδ )rδ αδ (1 − 2δ).
∗
Hence we have xδ , fδ − xδ ≥ (1 − αδ )rδ (1 − 2δ). Similarly, we also have x∗δ , fδ − yδ ≥
(1 − αδ − 2αδ δ)rδ . The above two inequalities imply that
(fδ − xδ ) + (fδ − yδ ) ≥ x∗δ , fδ − xδ + x∗δ , fδ − yδ ≥ 2rδ (1 − αδ − δ)
so that
fδ − xδ
fδ − yδ ≥ 2.
+
lim inf δ→0+ rδ (1 − αδ )
rδ (1 − αδ ) Note that
δ
fδ − xδ fδ − yδ H(Fαδδ , xδ )
≤ lim sup H(Fαδ , yδ ) ≤ 1.
≤ 1, lim sup
≤ lim sup
lim sup
rδ (1 − αδ )
rδ (1 − αδ )
δ→0+
δ→0+ rδ (1 − αδ )
δ→0+
δ→0+ rδ (1 − αδ )
Using the uniform convexity of X, we have that
xδ − yδ = 0,
lim
δ→0+ rδ (1 − αδ ) Li C. and Lopez G.
746
and so limδ→0 xδ − yδ = 0 as rδ (1 − αδ ) ≤ r0 /2. This contradicts that xδ − yδ > and the
proof is complete.
For F ∈ B, let Yδ (F ) = {x ∈ G : H(F, x) ≤ rG (F ) + δ}. It is obvious that the problem
(F, G) is well-posed if and only if limδ→0+ diamYδ (F ) = 0.
Lemma 3.2 Suppose that X is uniformly convex. Then the set B o (resp. K ∩ B o ) is a
dense Gδ -subset of B (resp. K ).
Proof We just prove Lemma 3.2 for B since the proof for K is similar. Let
1
.
Φn = F ∈ B : lim diamYδ (F ) ≥
δ→0+
n
Then
+∞
(B \ Φn ) .
Bo =
n=1
To complete the proof it suffices to prove that Φn is a closed subset of B and B \ Φn is dense
in B.
Suppose that Fk ∈ Φn , F0 ∈ B and H(Fk , F0 ) → 0. For any > 0, let k0 > 0 satisfy
H(Fk , F0 ) ≤ 3 , ∀ k > k0 . Then
rG (Fk ) ≤ rG (F0 ) + H(Fk , F0 ) ≤ rG (F0 ) + .
3
This implies
2
V Fk , rG (Fk ) +
⊂ V Fk , rG (F0 ) + ⊂ V (F0 , rG (F0 ) + ),
3
3
where V (F, r) = {x ∈ X : H(F, x) ≤ r}. Hence Y 3 (Fk ) ⊂ Y (F0 ) and
1
lim diamY (F0 ) ≥ lim diamY 3 (Fk ) ≥ .
→0
→0
n
This means that F0 ∈ Φn and so Φn is closed.
Now let us show that B \ Φn is dense in B. Let F ∈ Φn and 0 < α < 1/2 be arbitrary.
By Lemma 3.1 for any 0 < < 1/n, there exists δ0 > 0 such that, for any 0 < δ < δ0 and any
x ∈ G with r ≤ H(F, x) ≤ (1 + δ)r, we have diamMδ (F, x, r, α) < < n1 , where r = rG (F ). Let
x0 ∈ G with r ≤ H(F, x0 ) ≤ (1 + α2 δ)r and let Fα = (1 − α)F + αx0 . Then
α
H(Fα , F ) ≤ αH(F, x0 ) ≤ α 1 + δ r.
2
Clearly,
α
rG (Fα ) ≤ H(Fα , x0 ) ≤ (1 − α) 1 + δ r.
2
With ζ =
α
2 δr
the inclusion Yζ (Fα ) ⊂ Mδ (F, x0 , r, α) holds because for y ∈ Yζ (Fα ),
α
α
H(Fα , y) ≤ (1 − α) 1 + δ r + δr ≤ (1 − α)r + αδr.
2
2
Hence limδ→0+ diamYδ/2 (Fα ) < <
Lemma 3.3
1
n
and Fα ∈ B \ Φn . The proof is complete.
Let N = {1, 2, . . .}. Define
B̃ =
BH (Fα , ρFα (1/k)) ,
k∈N F ∈Bo 0≤α≤1/2
where Fα = (1 − α)F + αgF and ρFα () = min{H(F, Fα ), 1}. Then, B̃ ⊂ B o .
Proof It suffices to show that for every F ∈ B̃, limδ→0+ diamYδ (F ) = 0. Let F ∈ B̃ be
arbitrary. Then for each k ∈ N there exist F k ∈ B o and 0 ≤ αk ≤ 1/2 such that H(F, Fαkk ) ≤
Well-posedness of Chebyshev Center Problems
747
ρFαk (1/k), where Fαkk = (1 − αk )F k + αk gF k . This implies that H(F, Fαkk ) → 0 as k → +∞. It
k
follows that limk→+∞ rG (Fαkk ) = rG (F ). We claim that
rG (Fαkk ) = H(Fαkk , gF k ) = (1 − αk )rG (F k ).
In fact, it is evident that
rG (Fαkk ) ≤ H(Fαkk , gF k ) = (1 − αk )rG (F k ).
Now suppose that gk ∈ G is such that H(Fαkk , gk ) < (1 − αk )rG (F k ). Then
H(F k , gk ) ≤ H(Fαkk , F k ) + H(Fαkk , gk ) ≤ αk rG (F k ) + (1 − αk )rG (F k ) < rG (F k ),
which contradicts that gF k is the Chebyshev center of F k with respect to G and the claim is
proved. Let r0 = maxk∈N rG (F k ). Then r0 < +∞ since rG (Fαkk ) = (1 − αk )rG (F k ). Write
δk = ρFαk (1/k). Then, for all k large enough, we have Yδk (F ) ⊂ Y3δk (Fαkk ) since for any
k
g ∈ Yδk (F ),
H(Fαkk , g) ≤ H(Fαkk , F ) + H(F, g) ≤ rG (F ) + 2δk ≤ rG (Fαkk ) + 3δk .
Furthermore, observe that, for any g ∈ Y3δk (Fαkk ), H(Fαkk , g) ≤ (1 − αk )rk + αk rk (3/k), where
rk = rG (F k ) = H(F k , gF k ). It follows that Yδk (F ) ⊂ Y3δk (Fαkk ) ⊂ M3/k (F k , gFk , rk , αk ).
Thus, using Lemma 3.1, we have that limk→+∞ diamYδk (F ) = 0, thus proving the lemma.
Theorem 3.1 Let X be a uniformly convex Banach space. Then the set B\B o (resp. K \B o ,
BC \ B o , K C \ B o ) is σ-porous in B (resp. K , BC , K C ).
Proof We prove the theorem only for B as in the other cases the proof is similar. Let
1
BH (Fα , ρFα (1/k)) , Bkl = F ∈ Bk : < rG (F ) < l .
Bk = B \
l
o
F ∈B 0≤α≤1/2
Then, from Lemma 3.3, we have
B \ B o ⊂ B \ B̃ =
Bkl .
k∈N l∈N
To complete the proof it suffices to show that the set Bkl is porous in B for every k, l ∈ N .
Now let k, l ∈ N be arbitrary. Define r0 = 1/(2l) and α = min{1/(4k), 1/(r0 k)}. By
Lemma 3.2, for any F ∈ Bkl and 0 < r ≤ r0 , there exists F̄ ∈ B o such that H(F, F̄ ) < 4r
and 1l < rG (F̄ ) < l. Then, by the arguments as in the proof of Lemma 3.3, we get that, for
0 ≤ t ≤ 1,
rG (F̄t ) = H(F̄t , gF̄ ) = (1 − t)rG (F̄ ),
where F̄t = (1 − t)F + tgF̄ . Hence, for 0 ≤ t ≤ 1,
H(F̄t , F̄ ) ≥ H(F̄ , gF̄ ) − H(F̄t , gF̄ ) = rG (F̄ ) − rG (F̄t ) = t rG (F̄ ).
Consequently,
H(F̄1/2 , F ) ≥ H(F̄1/2 , F̄ ) − H(F̄ , F ) ≥ (1/2)rG (F̄ ) − r/4 ≥ 3r/4.
It follows that there exists 0 < t ≤ 1/2 such that F̄t satisfies H(F̄t , F ) = 3r/4. Since for each
A ∈ BH (F̄t , αr)
3
H(A, F ) ≤ H(A, F̄t ) + H(F̄t , F ) ≤ αr + r ≤ r,
4
we have that BH (F̄t , αr) ⊂ BH (F, r). In order to show that BH (F̄t , αr) ⊂ B \ Bkl , it suffices
to show that BH (F̄t , αr) ⊂ BH (F̄t , ρF̄t (1/k)). Indeed, from the definition of α, it follows that
αr ≤ 1/k. Furthermore, since H(F̄t , F̄ ) ≥ H(F̄t , F )−H(F, F̄ ) ≥ r2 , we have αr ≤ 2αH(F̄t , F̄ ) ≤
H(F̄t , F̄ ) k1 , so that αr ≤ min{H(F̄t , F̄ ), 1} k1 = ρF̄t (1/k). This completes the proof.
4 Ambiguous Loci
Recall that Ud (x, r) stands for the open ball with center x and radius r > 0. Then we can state
the following definition, see [5, 21]:
Li C. and Lopez G.
748
Definition 4.1 Let A be a subset of a metric space E with the metric d. A is called everywhere uncountable in E if, for every x ∈ E and r > 0, the set A ∩ Ud (x, r) is nonempty and
uncountable.
Clearly, if A ⊂ E is everywhere uncountable in E, then A is dense in E. The converse is
not true, in general.
Let KG denote the closure of the set {F ∈ K : rG (F ) > rX (F )} and let
A(G) = {F ∈ KG : (F, G) is not well-posed}.
Then the main theorem of this section can be stated as follows.
Theorem 4.1 Suppose that X is a separable strictly convex Banach space. Then the set
B ∗ = {G ∈ B : A(G) is everywhere uncountable in KG }
is residual in B.
Proof For F ∈ K and r > 0, define
BF,r = {G ∈ BF : A(G) ∩ UH (F, r) = ∅ or at most countable},
where BF denotes the closure of the set {G ∈ B : rG (F ) > rX (F )}. We will prove that BF,r
is nowhere dense in B.
Indeed, let G ∈ BF,r . Without loss of generality, we may assume that rG (F ) > rX (F )
(otherwise, we can take G̃, near G, such that G̃ has this property by the definition of BF ).
Furthermore, we also can assume that F ∈
/ A(G). In fact, suppose on the contrary that the
problem (F, G) is not well-posed. Note that rG (F ) > rX (F ). Let > 0 be arbitrary and
0 < δ < 12 (rG (F ) − rX (F )). Take g ∈ G such that H(F, g ) < rG (F ) + δ and g ∈ G such
that H(F, g ) < rX (F ) + δ. Then g − g ≤ b, where b = H(G, 0) + H(F, 0) + rG (F ). Set
gδ = (1 − )g + g and G̃ = G ∪ {gδ }. Then, H(G̃, G) ≤ dG (gδ ) ≤ gδ − g ≤ b. On the other
hand, gδ is the Chebyshev center of F with respect to G̃ and the problem (F, G̃) is well-posed
as
H(F, gδ ) ≤ (1 − )H(F, g ) + H(F, g ) ≤ (1 − )rG (F ) + δ + (rX (F ) + δ) < rG (F ).
Hence, we can assume that rG (F ) > rX (F ) and F ∈
/ A(G). Then, there exists a unique
gF ∈ G such that H(F, gF ) = rG (F ). Let 0 < < min{r, rG (F )} and take g1 = g2 ∈ X
such that 0 < gi − gF ≤ /4 for i = 1, 2 and H(F, g1 ) = H(F, g2 ) < H(F, gF ). Furthermore,
Fα = αF + (1 − α)gF . Since H(F0 , F ) = H(F, gF ) > H(F, gi ) and H(F1 , F ) = 0 < H(F, gi ),
i = 1, 2, there exists 0 < α < 1 such that H(Fα , F ) = H(F, gi ), i = 1, 2. Note that, for i = 1, 2,
H(F, gF ) ≤ H(F, gi ) + gi − gF = H(F, Fα ) + gi − gF .
We have that αrG (F ) = H(Fα , gF ) ≤ gi − gF ≤ /4, i = 1, 2, and α ≤ 4rG(F ) . It follows that
3
H(F, gi ) = H(F, Fα ) = (1 − α)rG (F ) ≥ rG (F ) − ≥ , i = 1, 2.
4
4
Let = min{, rG (F ) − H(F, g1 )}. Then H(F, gi ) ≤ rG (F ) − , i = 1, 2. Now define Y =
G ∪ {g1 , g2 }. We have that H(G, Y ) ≤ /4 because gi − gF ≤ /4. Set τ = 8H(F,g
) and
1
1
1
(H(F2δ , g1 ) − H(F2δ , g2 )), (H(F1δ , g2 ) − H(F1δ , g1 )) ,
h = min min
2
2
δ∈[τ /2,τ ]
δ
where Fi = (1 − δ)F + δgi , i = 1, 2. Then h > 0. In fact, otherwise, without loss of generality,
we may assume that H(F2δ , g1 ) ≤ H(F2δ , g2 ) for some δ ∈ [τ /2, τ ]. Take f ∈ F such that
f − g1 = H(F, g1 ). Then, using the strict convexity,
H(F, g1 )
= f − g1 = f − [(1 − δ)f + δg2 ] + [(1 − δ)f + δg2 ] − g1 < f − [(1 − δ)f + δg2 ] + [(1 − δ)f + δg2 ] − g1 ≤ δH(F, g2 ) + H(F2δ , g1 )
≤ δH(F, g2 ) + H(F2δ , g2 ) = H(F, g2 ),
Well-posedness of Chebyshev Center Problems
749
which contradicts that H(F, g2 ) = H(F, g1 ).
Define ρ = min{ 21 g1 − g2 , 4 , h}. For any Z ∈ UH (Y, ρ), define Zi = B(gi , ρ) ∩ Z, i = 1, 2.
Then Zi = ∅ for i = 1, 2 and Z1 ∩ Z2 = ∅.
Claim rZ (Aδt ) = rZ1 ∪Z2 (Aδt ), for 0 ≤ t ≤ 1 and τ /2 ≤ δ ≤ τ, where Aδt = (1 − δ)F + δ[(1 −
t)g1 + tg2 ].
First, we have that
rZi (Aδt ) ≤ H(Aδt , gi ) + ρ ≤ H(Aδt , Fiδ ) + H(Fiδ , gi ) + ρ
≤ δg1 − g2 + (1 − δ)H(F, gi ) + ρ
≤
g1 − g2 + 1 −
H(F, gi ) + ρ
8H(F, g1 )
16H(F, g1 )
15
+ ρ = rG (F ) − + ρ.
≤ + rG (F ) − −
8
16
16
On the other hand, rG (Aδt ) ≥ rG (F )−H(Aδt , F ) ≥ rG (F )− 8 , since H(Aδt , F ) ≤ δH(F, gi ) ≤ 8 .
Now, for any z ∈ Z with z−gi > ρ, since H(Y, Z) < ρ, there exists y ∈ Y such that z−y < ρ
so that y = gi for i = 1, 2 and y ∈ G. Thus,
H(Aδt , z) ≥ H(Aδt , y) − ρ ≥ rG (Aδt ) − ρ ≥ rG (F ) − − ρ.
8
+
ρ
≤
−
−
ρ.
This
completes
the
proof
of
the
claim.
Observe that − 15
16
8
On the other hand, from the definition of ρ, it follows that
rZ1 (F1δ ) ≤ H(F1δ , g1 ) + ρ
1
≤ H(F1δ , g1 ) + [H(F1δ , g2 ) − H(F1δ , g1 )]
2
1
δ
= H(F1 , g2 ) + [H(F1δ , g1 ) − H(F1δ , g2 )]
2
1
δ
≤ rZ2 (F1 ) + ρ + [H(F1δ , g1 ) − H(F1δ , g2 )]
2
≤ rZ2 (F1δ ).
A similar argument shows that rZ2 (F2δ ) ≤ rZ1 (F2δ ). For each δ ∈ [τ /2, τ ], let r(t) = rZ1 (Aδt ) −
rZ2 (Aδt ), for 0 ≤ t ≤ 1. Then r(0) = rZ1 (F1δ ) − rZ2 (F1δ ) ≤ 0, r(1) = rZ1 (F2δ ) − rZ2 (F2δ ) ≥ 0. It
follows from the continuity of r(·) that there exists 0 ≤ tδ ≤ 1 such that r(tδ ) = 0 so that
rZ1 (Aδtδ ) = rZ2 (Aδtδ ) = rZ (Aδtδ ),
for each δ ∈ [τ /2, τ ]. Since Z1 and Z2 are closed and disjoint, it follows that the problem
(Aδtδ , Z) is not well-posed for each δ ∈ [τ /2, τ ] and so Z ∈
/ BF,r . This implies that BF,r is
nowhere dense in B because Z ∈ UH (Y, ρ) is arbitrary.
It follows from the separability of X that K is separable. Let S be a countable dense
subset of K and let Q+ denote the set of all positive rationals. Define
B̃ =
BF,r .
F ∈S r∈Q+
Then B̃ is of the first Baire category in B. Hence, it remains to verify that B ∗ ⊇ B \ B̃.
Let G ∈ B \ B̃. For any F ∈ KG (X) and any r > 0, there exist F̂ ∈ S and r̂ ∈ Q+ such
that rG (F̂ ) > rX (F̂ ) and UH (F̂ , r̂) ⊆ UH (F, r). Hence, F̂ ∈ BF̂ and A (G) ∩ UH (F̂ , r̂) is
nonempty and uncountable since G ∈
/ BF̂ ,r̂ . Consequently, A (G) ∩ UH (F, r) is nonempty and
uncountable and G ∈ B ∗ . This completes the proof of Theorem 4.1.
References
[1] Dontchev, A., Zolezzi, T.: Well Posed Optimization Problems, Lecture Notes in Math., 1543, SpringerVerlag, Berlin, Heidelberg, 1993
750
Li C. and Lopez G.
[2] Borwein, J. M., Fitzpatrick, S.: Existence of nearest points in Banach spaces. Can. J. Math., 41, 702–720
(1989)
[3] De Blasi, F. S., Myjak, J., Papini, P. L.: Porous sets in best approximation theory. J. London Math. Soc.,
44(2), 135–142 (1991)
[4] De Blasi, F. S., Myjak, J.: On a generalized best approximation problem. J. Approx. Theory, 94, 54–72
(1998)
[5] De Blasi, F. S., Myjak, J.: Ambiguous loci in best approximation theory, in “Approximation Theory, Spline
Functions and Applications” (S. P. Singh ed.), Kluwer Acad. Publ., Dordrecht, 341–349, 1992
[6] De Blasi, F. S., Myjak, J., Papini, P. L.: On mutually nearest and mutually furthest points of sets in Banach
spaces. J. Approx. Theory, 70, 142–155 (1992)
[7] Li, C.: On mutually nearest and mutually furthest points in reflexive Banach spaces. J. Approx. Theory,
103, 1–17 (2000)
[8] Garkavi, A. L.: The best possible net and the best possible cross-section of a set in a normed space. Izv.
Akad. Nauk. USSR, 26, 87–106 (1962)
[9] Beer, G., Pai, D. V.: On convergence of convex sets and relative Chebyshev centers. J. Approx. Theory,
62, 147–169 (1990)
[10] Li, C., Wang, X. H.: Lipschitz continuity of best approximations and Chebyshev centers. Chinese Science
Bull., 43, 185–189 (1998)
[11] Li, C., Wang, X. H.: Almost Chebyshev set with respect to bounded subsets. Science in China, 40, 375–383
(1997)
[12] Pai, D. V., Nowroji, P. T.: On restricted centers of sets. J. Approx. Theory, 66, 170–189 (1991)
[13] Steckin, S. B.: Approximation properties of sets in normed linear spaces. Rev. Roumaine Math. Pures and
Appl., 8, 5–18 (1963)
[14] Deville, R., Revalski, J. P.: Porosity of ill-posed problems. Proc. Amer. Math. Soc., 128, 1117–1124 (2000)
[15] Georgiev, P. G.: The strong Ekeland variational principle, the strong drop theorem and applications. J.
Math. Anal. Appl., 131, 1–21 (1988)
[16] Kenderov, P. S., Revalski, J. P.: Generic well-posedness of optimization problems and the Banach–Mazur
game, in Recent Developments in Well-posed Variational Problems ( Lucchetti–Revalski, eds), Mathematics
and Applications, Kluwer Acad. Publ., Dordrecht, 331, 117–136, 1995
[17] Lau, K. S.: Almost Chebyshev subsets in reflexive Banach spaces. Indiana Univ. Math. J., 27, 791–795
(1978)
[18] Li, C.: Almost K-Chebyshev sets. Act. Math. Sinica, Chinese Series, 33(2), 251–259, (1990)
[19] Li, C.: On well posedness of best simultaneous approximation problems in Banach spaces. Science in China
(Ser. A), 44, 1558–1570 (2001)
[20] Shunmugaraj, P.: Well-set and well-posed minimization problems. Set-valued Analysis, 3, 281–294 (1995)
[21] De Blasi, F. S., Myjak, J.: Ambiguous loci of the nearest point mapping in Banach spaces. Arch. Math.,
61, 377–384 (1993)
[22] Zhivkov, N. V.: Compacta with dense ambiguous loci of metric projections and antiprojections. Proc.
Amer. Math. Soc., 123, 3403–3411 (1995)
[23] Pheips, R. R.: Convex Functions, Monotone Operators and Differentiability, Lect Note in Math., 1364,
Springer-Verlag, New York, 1989
[24] Preiss, D.: Differentiability of Lipschitz functions on Banach spaces. J. Funct. Anal., 91, 312–345 (1990)
[25] Amir, D., Ziegler, Z.: Relative Chebyshev centers in normed linear spaces. J. Approx. Theory, 29, 235–252
(1980)
[26] Diestel, J.: Geometry of Banach Spaces–selected Topic, Lect Notes in Math, 485, Spring-Verlag, New York,
1975
[27] Radstrom, H.: An embedding theorem for spaces of convex sets. Proc. Amer. Math. Soc., 3, 165–169
(1952)