C O L L O Q U I U M M A T H E M A T I C U M VOL. 81 1999 NO. 2

VOL. 81 1999 NO. 2

SOME STRUCTURES RELATED

TO METRIC PROJECTIONS IN ORLICZ SPACES

BY

BOR-LUH L I N (IOWA CITY, IA) ^AND ZHONGRUI S H I (IOWA CITY, IA,ANDHARBIN)

Abstract. We discuss k-rotundity, weak k-rotundity, C-k-rotundity, weak C-k-rotun- dity, k-nearly uniform convexity, k-β property, C-I property, C-II property, C-III property and nearly uniform convexity both pointwise and global in Orlicz function spaces equipped with Luxemburg norm. Applications to continuity for the metric projection at a given point are given in Orlicz function spaces with Luxemburg norm.

Let X be a Banach space, and D be a subset of X. The metric projection P

: X → 2

is defined by P

(x) = {y ∈ D : kx − yk = dist(x, D)}. D is a proximinal (resp. Chebyshev ) set if P

(x) contains at least (resp. exactly) one point for all x in X. For a proximinal D, P

is called norm-norm (resp.

norm-weak ) upper semicontinuous at x if for every normed (resp. weak) open set W ⊇ P

(x), there exists a normed neighborhood U of x such that P

(y) ⊆ W for all y in U . It is proved in [Wa95] that if X has the C-II (or C-III) property, then P

is continuous for any Chebyshev convex set D.

In this paper, we investigate some structures which imply the continuity of the metric projection at a given point for Orlicz function spaces with Luxemburg norm.

Let B(X) and S(X) be the unit ball and the unit sphere of the Banach space X respectively. A point x ∈ S(X) is said to be a locally C-I (resp.

C-II, C-III) point of B(X) if the following implication holds for every sequence {x

} ⊆ B(X): if for any δ > 0 there exists an integer m such that conv({x} ∪ {x

}

) ∩ (1 − δ)B(X) = ∅, then lim

x

= x (resp. {x

} is relatively compact, weakly compact) [Wa95]. We call such points LC-I, LC-II, and LC-III points respectively.

1991 Mathematics Subject Classification: 46B20, 46E30.

Key words and phrases: k-rotundity, weak k-rotundity, C-k-rotundity, weak C-k- rotundity, k-nearly uniform convexity, k-β property, C-I property, C-II property, C-III property, locally uniform convexity, nearly uniform convexity, Luxemburg norm, Orlicz function spaces.

The work was supported in part by the NSF and JYF of China.

[223]

Recall that the Kuratowski measure of noncompactness α(A) for A ⊂ X is defined as

α(A) = inf{ε > 0 : A can be covered by a finite family of sets

of diameter less than ε}.

A slice of B(X) is defined by S(f, η) = {x ∈ B(X) : f (x) > 1 − η} where f ∈ S(X

) and η > 0.

Let R be the set of all real numbers. A function M : R → R

is called an Orlicz function if M is convex, even, M (0) = 0 and M (∞) = ∞. The complementary function N of M in the sense of Young is defined by

N (v) = sup

u∈R

{uv − M (u)}.

It is known that if M is an Orlicz function, then so is N . M is said to be strictly convex if M ((u + v)/2) < (M (u) + M (v))/2 for all u 6= v. An interval (a, b) is said to be an affine interval of M if M is affine on (a, b) and M is strictly convex on (b, b + ε) and (a − ε, a) for some ε > 0. Denote all affine intervals of M by S

i=1

(a

, b

).

M is said to satisfy the △

-condition for large u (we simply write M∈ △

) if for some K and u

> 0, M (2u) ≤ KM (u) for |u| ≥ u

.

Let G be a bounded set in R

and let (G, Σ, µ) be a finite non-atomic measure space. For a real-valued measurable function x(t) over G, we call

̺

(x) =

T

G

M (x(t)) dµ(t) the modular of x. The Orlicz function space L

generated by M is the Banach space

L

= {x = x(t) : ∃λ > 0, ̺

(λx) < ∞}

equipped with the Luxemburg norm

kxk = inf{λ : ̺

(x/λ) ≤ 1}.

For information on Orlicz spaces, see [KrRu61, Ch96].

First we recall some lemmas.

Lemma 1 [LiSh96]. In an Orlicz function space L

equipped with Lux- emburg norm , let x ∈ S(L

). If M does not satisfy the △

-condition, then α(S(f, η)) ≥ 1/4 for any slice S(f, η) of B(L

) containing x.

Lemma 2 [LiSh96]. In an Orlicz function space L

equipped with Lux- emburg norm, let x ∈ S(L

). If µ{t ∈ G : x(t) ∈ S

i=1

(a

, b

)} > 0, where S

i=1

(a

, b

) is the family of all affine intervals of M , then α(S(f, η)) ≥ θ > 0

for any slice S(f, η) of B(L

) containing x, where θ is a constant that

depends only on x.

Theorem 1. In an Orlicz function space L

equipped with Luxemburg norm, let x ∈ S(L

). Then the following are equivalent:

(1) x is an LC-II point of B(L

).

(2) (i) M ∈ △

,

(ii) µ{t ∈ G : |x(t)| ∈ S

i=1

(a

, b

)} = 0, where S

i=1

(a

, b

) is all affine intervals of M ,

(iii) if µ{t ∈ G : |x(t)| = b} > 0 for some affine interval (a, b), then N ∈ △

and µ{t ∈ G : |x(t)| = c} = 0 for all affine intervals (c, d) of M .

(3) x is an LUR point of B(L

), i.e., for all sequences {x

} in B(L

), lim

kx

− xk = 0 whenever lim

kx

+ xk = 2.

P r o o f. (1)⇒(2). (i) Suppose that M 6∈ △

. Then (see the proof of Lemma 1 in [LiSh96]) there is a sequence {x

} satisfying

x

= x|

+ (x + u

)|

, lim

n→∞

kx

k

= 1, α({x

}) ≥ 1/4, and x

→ x weakly. For every δ > 0 there exists an integer N so that conv({x} ∪ {x

}

) ∩ (1 − δ)B(X) = ∅; but α({x

}) ≥ 1/4, which contra- dicts x being an LC-II point of B(L

).

(ii) Suppose µ{t ∈ G : |x(t)| ∈ S

i=1

(a

, b

)} > 0. By Lemma 2, there exists a sequence {x

} in B(L

) satisfying α({x

}) ≥ θ and x

→ x weakly, where θ depends only on x, which implies that x is not an LC-II point of B(L

), a contradiction.

(iii) Suppose that µB = µ{t ∈ G : |x(t)| = b} > 0 and µC = µ{t ∈ G :

|x(t)| = c} > 0 for some affine intervals (a, b) and (c, d) of M . Take B

⊂ B and C

⊂ C with µB

> 0, µC

> 0 and

[M (b) − M (a)]µB

= [M (d) − M (c)]µC

(i.e., M (b)µB

+ M (c)µC

= M (a)µB

+ M (d)µC

). Set

z = x|

+ a + b

2 sign x|

+ c + d

2 sign x|

. Then

̺

(z) = ̺

(x|

) + M (a) + M (b)

2 µB

+ M (c) + M (d)

2 µC

= ̺

(x) = 1.

As in the proof of Lemma 2, there exists a sequence {z

} in B(L

) sat- isfying α({z

}) ≥ θ and z

→ z weakly, where θ depends only on z, hence only on x. Let y = x|

+ a sign x|

+ d sign x|

. Then

̺

(y) = ̺

(x|

) + M (a)µB

+ M (d)µC

= ̺

(x) = 1

and z = (x + y)/2. Since kxk

= kyk

= kzk

= 1, there is f ∈ L

with f (x) = f (z) = kf k = 1. Since z

→ z weakly, for any δ > 0 there exists an integer N so that conv({x} ∪ {z

}

) ∩ (1 − δ)B(X) = ∅; but α({z

}) ≥ θ contradicts x being an LC-II point of B(L

).

Suppose µB = µ{t ∈ G : |x(t)| = b} > 0 for some affine interval (a, b) of M and N 6∈ △

. Since N 6∈ △

, there exist u

ր ∞ such that

2

M  1 2

u



>

 1 − 1

n



M (u

).

Without loss of generality, assume that x(t) = b on B. Take subsets B

in B such that B ⊃ B

⊃ B

⊃ . . . and

[M (u

) − M (a)]µB

= [M (b) − M (a)]µB.

Then M (u

)µB

≥ [M (b) − M (a)]µB. Set

x

= x|

+ a|

+ u

|

. Then

̺

(x

) = ̺

(x|

) + M (a)(µB − µB

) + M (u

)µB

= ̺

(x|

) + M (b)µB = ̺

(x) = 1.

Obviously

β→0

lim sup

n

̺

(βx

)

β ≥ [M (b) − M (a)]µB > 0,

by [An62], {x

} is not weakly compact and so α({x

}) ≥ θ > 0. For any δ > 0, take K > 0 such that 2/K ≤ δ. Set x

= x. Then for all K < n

<

. . . < n

and any P

i=0

λ

= 1, λ

≥ 0, we have

̺

 X

i=0

λ

x

 = ̺

(x|

) + M  λ

b +

k

X

i=1

λ

a 

µ(B \ B

)

+ M  X

i=1

λ

u

+ λ

b  µB

+ M 

X

i=1

λ

u

+ λ

b + λ

a  



≥ ̺

(x|

) + 

λ

M (b) +

k

X

i=1

λ

M (a) 

µ(B \ B

)

+

k

X

i=1, λi≥1/2ⁿⁱ

(1 − 1/n

)λ

M (u

)µB

+ M (λ

b)µB

+ M 

X

i=1

λ

u

+ λ

b + λ

a 

µ(B

\ B

)

+ M 

X

i=1

λ

u

+ λ

b + λ

a  



≥ ̺

(x|

) + 

λ

M (b) +

k

X

i=1

λ

M (a) 

µ(B \ B

)

+

k

X

i=1, λi≥1/2ⁿⁱ

(1 − 1/n

)λ

M (u

)µB

+ M (λ

b)µB

+

k−1

X

i=1, λi≥1/2ⁿⁱ

(1 − 1/n

)λ

M (u

)µ(B

\ B

) + M (λ

b + λ

a)µ(B

\ B

)

+ M 

X

i=1

λ

u

+ λ

b + λ

a  



≥ ̺

(x|

) + 

λ

M (b) +

k

X

i=1

λ

M (a) 

µ(B \ B

)

+

k

X

j=1 j

X

i=1, λi≥1/2ⁿⁱ

(1 − 1/n

)λ

M (u

)µ(B

\ B

)

+

k

X

j=1

M (λ

b + (λ

+ . . . + λ

)a)µ(B

\ B

)

≥ ̺

(x|

) + 

λ

M (b) +

k

X

i=1

λ

M (a) 

µ(B \ B

)

+ (1 − 1/n

)

k

X

j=1 j

X

i=1, λi≥1/2ⁿⁱ

λ

M (u

)µ(B

\ B

)

+

k

X

j=1

M (λ

b + (λ

+ . . . + λ

)a)µ(B

\ B

)

≥ (1 − 1/n

)

k

X

i=1

λ

̺

(x

) − (1 − 1/n

)

k

X

i=1, λi<1/2ni

λ

̺

(x

)

≥ (1 − 1/n

) −

k

X

i=1

1/2

= (1 − 1/K) − 1/2

> 1 − δ.

Hence conv({x}∪ {x

}

)∩ (1− δ)B(X) = ∅; but α({x

}) > 0 contradicts x being an LC-II point of B(L

).

(2)⇒(3). By [ChWa92], it follows that x is an LUR point of B(L

).

(3)⇒(1). Obvious.

For an integer k, a point x ∈ S(X) is said to be:

• a locally k-rotund (LkR) point of B(X) if for any sequence {x

} in B(X), lim

kx+x

+. . .+x

k = k+1 implies lim

kx

−xk = 0;

• a locally weakly k-rotund (LWkR) point of B(X) if for any sequence {x

} in B(X), lim

kx + x

+ . . . + x

k = k + 1 implies w-lim

x

= x;

• a locally C-k-rotund (LCkR) point of B(X) if for any sequence {x

} in B(X), lim

kx + x

+ . . . + x

k = k + 1 implies {x

} is a relatively compact set;

• a locally k-nearly uniformly convex (LkNUC) point of B(X) if for every ε > 0 there exists δ > 0 such that for all sequences {x

} with sep(x

) ≥ ε there are {n

, . . . , n

} with

x + x

+ . . . + x

k + 1

≤ 1 − δ;

• a locally k-β (Lkβ) point of B(X) if for every ε > 0 there exists δ > 0 such that for all sequences {x

} with sep(x

) ≥ ε there are {n

, . . . , n

} with conv({x, x

, . . . , x

}) ∩ (1 − δ)B(X) 6= ∅;

• a locally nearly uniformly convex (LNUC) point of B(X) if for every ε > 0 there exists δ > 0 such that for all sequences {x

} with sep(x

) ≥ ε we have conv({x} ∪ {x

}) ∩ (1 − δ)B(X) 6= ∅.

It is easy to see that for all Banach spaces, we have the implications

LUR LkR LCkR

LWkR LC-II

LkNUC Lkβ LNUC

+3

_____

_____



+3





+3

___

___ ^______

+3

KS

For these properties, we refer to [Ku91, KuLi94, KuLi93, Wa95].

Corollary 1. In an Orlicz function space L

equipped with Luxem- burg norm , let x ∈ S(L

). Then the following are equivalent:

(1) x is an LUR point of B(L

) [ChWa92];

(2) x is an LkR point of B(L

) (k ≥ 1);

(3) x is an LWkR point of B(L

) (k ≥ 1);

(4) x is an LCkR point of B(L

) (k ≥ 1);

(5) x is an LkNUC point of B(L

) (k ≥ 1);

(6) x is an Lk-β point of B(L

) (k ≥ 1);

(7) x is an LNUC point of B(L

);

(8) x is an LC-I point of B(L

);

(9) x is an LC-II point of B(L

);

(10) M ∈ △

, µ{t ∈ G : |x(t)| ∈ S

i=1

(a

, b

)} = 0, where {(a

, b

)} is the family of all affine intervals of M , and if µ{t ∈ G : |x(t)| = b} > 0 for some affine interval (a, b) of M , then N ∈ △

and µ{t ∈ G : |x(t)| = c} = 0 for all affine intervals (c, d) of M .

P r o o f. (1)⇒(2)⇒(3), (1)⇒(2)⇒(4)⇒(9), (1)⇒(5)⇒(6)⇒(7), and (1)⇒(8)⇒(9) are trivial by definitions.

(7)⇒(9). By Theorem 4 of [Wa95], an LNUC point is an LC-II point in B(X).

(10)⇒(1). This is proved in [ChWa92].

(9)⇒(10). This follows from Theorem 1.

(3)⇒(10). Since kxk

= 1, there is c > 0 such that µG

= µ{t ∈ G :

|x(t)| ≤ c} > 0.

Suppose that M 6∈ △

. Then there exist u

ր ∞ such that M ((1 + 1/n)u

) > 2

M (u

).

On passing to a subsequence if necessary, there are disjoint subsets G

⊂ G

so that

M (u

)µG

= 1/2

, n = 1, 2, . . . Define y = P

n=1

u

|

. Then ̺

(y) = P

n=1

M (u

)µG

= 1, kyk

= 1 and dist(y, E

) = 1, where E

= {x : ̺

(λx) < ∞ for all λ}. By the Hahn–Banach theorem, there is a functional φ such that φ(y) = kφk = 1, and φ(z) = 0 for all z in E

. Set x

= x|

+ y|

. Then

x + x

+ . . . + x

k + 1

≥ kx|

i>nkGi

k

→ 1 (n

, . . . , n

→ ∞) and

̺

(x

) = ̺

(x|

i>nGi

) + ̺

(y|

) → ̺

(x) ≤ 1.

But

φ(x

− x) = φ(y|

) − φ(x|

i>nGi

) = φ(y|

i>nGi

)

= φ(y|

) = 1.

So x

6→ x weakly, contrary to x being an LWkR point of B(L

).

We claim that µ{t ∈ G : |x(t)| ∈ S

i=1

(a

, b

)} = 0.

In fact, if this measure is positive, then µE > 0, where E = µ{t ∈ G :

x(t) ∈ (a + 2δ, b − 2δ)} for some δ > 0. Split E into two parts E

and E

with µE

= µE

= (µE)/2. Define

z = x|

+ (x + 2δ)|

+ (x − 2δ)|

. Then

̺

(z) = ̺

(x|

) + ̺

((x + 2δ)|

) + ̺

((x − 2δ)|

)

= ̺

(x|

) + ̺

(x|

) + ̺

(x|

) = 1,

̺

 x + z 2



= ̺

(x|

) + ̺

((x + δ)|

) + ̺

((x − δ)|

)

= ̺

(x|

) + ̺

(x|

) + ̺

(x|

) = 1.

Moreover x 6= z. As in Lemma 2, there exists a sequence {z

} in B(L

) such that z

→ z weakly and sep{z

} ≥ θ > 0, where θ depends only on z. For k > 1, since z

→ z weakly and kx + zk

= 2, we have lim

kx + z

+ . . . + z

k = k + 1. This contradicts x being an LWkR point of B(L

). For k = 1 we can take x

= z to get a contradic- tion.

From Theorem 1, it follows that if µ{t ∈ G : |x(t)| = b} > 0 for some affine interval (a, b) of M , then N ∈ △

and µ{t ∈ G : |x(t)| = c} = 0 for all affine intervals (c, d) of M .

Corollary 2. In an Orlicz function space L

equipped with Luxem- burg norm , the following are equivalent:

(1) L

is locally UR [ChWa92, Ka84];

(2) L

is locally kR (k ≥ 1);

(3) L

is locally WkR (k ≥ 1);

(4) L

is locally CkR (k ≥ 1);

(5) L

is locally kNUC (k ≥ 1);

(6) L

is locally k-β (k ≥ 1);

(7) L

is locally NUC ; (8) L

has the C-I property ; (9) L

has the C-II property ;

(10) M ∈ △

and M is strictly convex on the real line.

Corollary 3. In an Orlicz function space L

equipped with Luxem- burg norm , suppose M ∈ △

and let x ∈ S(L

). If µ{t ∈ G : |x(t)| ∈ S

i=1

(a

, b

)} = 0 and either µ{t ∈ G : |x(t)| ∈ S

i=1

{b

}} = 0, or N ∈ △

and µ{t ∈ G : |x(t)| ∈ S

i=1

{a

}} = 0, then every proximinal metric projec- tion P

is norm-norm upper semicontinuous at x.

Moreover , if M ∈ △

and M ∈ SC, then every proximinal metric projection P

is norm-norm upper semicontinuous.

Next, we study the LC-III points.

Lemma 3. For an Orlicz space L

, suppose M ∈ △

. Then (1) for any ε > 0 there is η > 0 such that

̺

(x) < η ⇒ kxk

< ε, kxk

> 1 − η ⇒ ̺

(x) > 1 − ε;

(2) if ̺

(x

) → ̺

(x) and x

→ x in measure, then x

→ x in norm.

For a proof, see [Ch86, Hu83, HuLa95].

Theorem 2. In an Orlicz function space L

equipped with Luxemburg norm, let x ∈ S(L

). Then x is a C-III point of B(L

) if and only if

(1) M ∈ △

;

(2) either N ∈ △

, or µ{t ∈ G : |x(t)| ∈ S

i=1

(a

, b

)} = 0 and µ{t ∈ G :

|x(t)| ∈ S

i=1

{b

}} = 0.

P r o o f. Choose c > 0 such that µG

= µ{t ∈ G : |x(t)| ≤ c} > 0. Sup- pose M 6∈ △

. There exists [KrRu61] y ∈ L

with supp y ⊂ G

, kyk

= dist(y, E

) = 1, and φ ∈ L

with φ(y) = kφk = dist(y, E

) = 1 and φ(z) = 0 for all z ∈ E

, and G

⊂ G

, where G

= {t ∈ G : |y(t)| ≥ n}. Set

x

= x|

+ y|

.

Then for θ > 0, take n

such that kx|

k

> 1 − θ. Then for all n

< n

< . . . < n

and for any P

i=0

λ

= 1, where λ

≥ 0,

k

X

i=0

λ

x

≥ kx|

k

> 1 − θ.

But {x

} is not relatively weakly compact. In fact, otherwise by the Shmul’yan Theorem {x

} is relatively weakly sequentially compact. By taking a subsequence if necessary we may assume that x

→ x

in the weak topology. Combining this with x

→ x in the w

topology, we get x

→ x.

A contradiction since φ(x

− x) = φ(y|

) + φ(x|

) = φ(y|

) = 1.

Assume that µ{t ∈ G : |x(t)| ∈ S

i=1

(a

, b

)} > 0. Then µB = µ{t ∈ G : x(t) ∈ (a + θ, b − θ)} > 0 for some affine interval (a, b) and some θ > 0. Split B into two parts B

, B

with µB

= µB

= (µB)/2. Define

y = x|

+ (x − θ)|

+ (x + θ)|

. Then

̺

(y) = ̺

(x|

) + ̺

((x − θ)|

) + ̺

((x + θ)|

)

= ̺

(x|

) + ̺

(x|

) + ̺

(x|

) = 1, and

̺

 x + y 2



= ̺

(x) = 1.

If N 6∈ △

, then there exists a real sequence {u

} such that u

ր ∞ and

2

M  1 2

u



>

 1 − 1

n



M (u

).

Take decreasing subsets {B

} of B such that

̺

(y|

) − M (a)µB = ̺

(x|

) − M (a)µB = [M (u

) − M (a)]µB

. Then M (u

)µB

≥ ̺

(x|

) − M (a)µB > 0. Set

x

= x|

+ a|

+ u

|

. By [An62], {x

} is not weakly compact. But

̺

(x

) = ̺

(x|

) + M (a)(µB − µB

) + M (u

)µB

= ̺

(x) = 1.

For any δ > 0, take K such that 2/K ≤ δ. Let x

= x. Then for all K < n

< . . . < n

and for any P

i=0

λ

= 1, where λ

≥ 0, as in the proof of Theorem 1,

̺

 X

i=0

λ

x



≥ 1 − δ.

This contradicts x being a C-III point of B(L

).

By the same argument as for the second part of (iii) in Theorem 1 we can show that if x is a locally C-III point of B(L

) then µ{t ∈ G : |x(t)|

= b} > 0 for some affine interval (a, b) of M implies N ∈ △

.

Suppose {x

} is a sequence in B(L

) such that for any δ > 0 there exists an integer N with conv({x} ∪ {x

}

) ∩ (1 − δ)B(L

) = ∅.

If N ∈ △

, then by (1), L

is reflexive. So B(L

) is weakly compact and {x

} is relatively weakly compact.

If N 6∈ △

, then we show that lim

x

= x. By Lemma 3, it suffices to show that x

→ x in measure. By (2), µ{t ∈ G : |x(t)| ∈

S

i=1

(a

, b

)} = 0 and µ{t ∈ G : |x(t)| = b} = 0 for all affine intervals (a, b). Since lim

kx+ x

+ . . .+ x

k

= k + 1, we have lim

kx+ x

k

= 2. From

1 = ̺

(x) + ̺

(x

)

2 ≥ ̺

 x + x

2



→ 1, it follows that x

→ x in measure on {t ∈ G : |x(t)| 6∈ G \

S

i=1

[a

, b

]}.

We claim: x

→ x in measure on G

= {t ∈ G : |x(t)| = a} for every left endpoint a of an affine interval (a, b). Without loss of generality, assume that G

= {t ∈ G : x(t) = a}.

We first show that for any ε > 0, µ{t ∈ G

: x

(t) ≤ a − ε} → 0 as

n → ∞. Indeed, if for some ε

> 0 and σ

> 0 and a subsequence of {x

}

(again denoted by {x

}) we have µG

= µ{t ∈ G

: x

(t) ≤ a−ε

} ≥ σ

> 0

for all n, then there exists a δ

> 0 such that M  a + a − ε

2



≤ 1

2 (1 − δ

)[M (a) + M (a − ε

)]

(because c 6= d for all affine intervals (c, d)). Hence

̺

 x + x

2



≤

[̺

(x|

) + ̺

(x

|

)] + M  a + a − ε

2

 µG

≤

[̺

(x|

) + ̺

(x

|

)]

+

(1 − δ

)[M (a) + M (a − ε

)]µG

≤

[̺

(x) + ̺

(x

)] −

δ

[M (a) + M (a − ε

)]µG

≤ 1 −

δ

[M (a) + M (a − ε

)]µG

< 1.

By Lemma 3, lim

kx + x

k

< 2, a contradiction.

Next we show that for any ε > 0, µ{t ∈ G

: x

(t) ≥ a + ε} → 0 as n → ∞. Indeed, suppose that for some ε

> 0 and σ

> 0 and a subsequence {x

} (again labeled {x

}) we have µG

= µ{t ∈ G

: x

(t) ≥ a + ε

} ≥ σ

for all n. Since G = n

t ∈ G : |x(t)| 6∈

∞

[

i=1

[a

, b

] o

∪ n

t ∈ G : |x(t)| ∈

∞

[

i=1

(a

, b

) o

∪ n

t ∈ G : |x(t)| ∈

∞

[

i=1

{b

} o

∪ n

t ∈ G : |x(t)| ∈

∞

[

i=1

{a

} o , by the Fatou Lemma, we see that for all G

⊂ G,

lim inf

n→∞

̺

(x

|

) ≥ ̺

(x|

).

Hence for n large enough,

̺

(x

) = ̺

(x

|

) + ̺

(x

|

)

≥ ̺

(x

|

) + M (a + ε

)µG

= ̺

(x

|

) + M (a)µG

+ [M (a + ε

) − M (a)]µG

≥ ̺

(x) + [M (a + ε

) − M (a)]σ

> 1, a contradiction.

We now show that x

→ x in measure on {t ∈ G : |x(t)| ∈

S

i=1

{a

}}.

Indeed, for every ε > 0 and σ > 0, take i

such that µ{t ∈ G : |x(t)| ∈ S

i>i0

{a

}} < ε/2. From the claim we deduce that for n large enough, µ n

t ∈ G : |x(t)| ∈

i0

[

i=1

{a

} and |x

(t) − x(t)| ≥ σ o

< ε 2 . From the decomposition of G as above we get x

→ x in measure on G.

By Lemma 3, we know that x

→ x in norm, so {x

} is relatively weakly compact.

Remark . By the same argument we can show that an element in S(L

) is a locally C-III point of B(L

) iff it is a locally WCkR point of B(L

).

Corollary 4. In an Orlicz function space L

equipped with Luxem- burg norm , the following are equivalent:

(1) L

is locally WCkR;

(2) L

has the C-III property;

(3) M ∈ △

and either M ∈ SC or N ∈ △

.

Corollary 5. In an Orlicz function space L

equipped with Luxem- burg norm, suppose M ∈ △

and let x ∈ S(L

). If µ{t ∈ G : |x(t)| ∈ S

i=1

(a

, b

)} = 0 and µ{t ∈ G : |x(t)| ∈ S

i=1

{b

}} = 0, then every proxim- inal metric projection P

is norm-weak upper semicontinuous at x.

Moreover , if M ∈ △

, and either M ∈ SC or N ∈ △

, then every prox- iminal metric projection P

is norm-weak upper semicontinuous on L

.

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Department of Mathematics The University of Iowa Iowa City, IA 52242, U.S.A.

E-mail: bill@math.uiowa.edu

Current address of Z. R. Shi:

Department of Mathematics Harbin University of Science and Technology 150080 Harbin, China E-mail: zshi@public.hr.hl.cn

Received 24 November 1998;

revised 15 February 1999