Some Generalization of Locally Uniform Rotundity - Library(ISI ...

Journal of Mathematical Analysis and Applications 252, 906916 Ž 2000.
doi:10.1006jmaa.2000.7169, available online at http:www.idealibrary.com on
Some Generalizations of Locally Uniform Rotundity
P. Bandyopadhyay
Stat-Math Diision, Indian Statistical Institute, 203, B. T. Road, Calcutta 700 035,
India
E-mail: pradipta@isical.ac.in
Da Huang and Bor-Luh Lin
Department of Mathematics, Uniersity of Iowa, Iowa City, Iowa 52242
E-mail: bor-luh-lin@uiowa.edu
and
S. L. Troyanski
Department of Mathematics, Uniersity of Sofia, 5 James Bourchier Bouleard,
Sofia, Bulgaria; and Departamento de Matematicas, ´ Uniersidad de Murcia,
Campus de Espinardo, 30100 Espinardo, Murcia, Spain
E-mail: slt@fmi.uni-sofia.bg, stroya@fcu.um.es
Submitted by R. M. Aron
Received January 12, 2000
Some new generalizations of locally uniform rotundity in Banach spaces are
introduced and studied. 2000 Academic Press
Key Words: almost locally uniformly rotund points; weakly almost locally uniformly
rotund points; rotund points; -fragmentable Banach spaces.
Recently P. Kenderov and W. Moors 9, 10
proved that every weakly
locally uniformly rotund Ž WLUR.
Banach space is -fragmentable in the
sense of J. Jayne et al. 8 . This result was generalized in 15
where it is
proved that every WLUR Banach space is locally uniformly rotund Ž LUR.
renormable. However, a careful analysis of 5, 9, 10 Žsee also 11.
shows
that a Banach space X is -fragmentable if X has the property that
x x** whenever x X, x** X**, and x x** Ž x x** . 2 .
It is easy to see that every WLUR Banach space has this property 17 . In
0022-247X00 $35.00
Copyright 2000 by Academic Press
All rights of reproduction in any form reserved.
906

LOCALLY UNIFORM ROTUNDITY 907
5, p. 420
the question of how to characterize this property is raised. In
this paper, we show that this property is strictly weaker than WLUR. We
characterize the property in terms of the initial space X without referring
to the bidual of X and in terms of unbounded nested sequence of balls in
the dual space X*.
DEFINITION 1. An ‘‘unbounded nested sequence of balls’’ in a Banach
space X is an increasing sequence B BŽ x , r .4
n n n of open balls in X
with rn.
DEFINITION 2. Let x be a point of the unit sphere SX
of a Banach
space X. We say that x is
Ž. a 4 a rotund point of the unit ball B of X Ž or, X is rotund at x.
X
if y Ž x y. 2
1 implies x y;
Ž b. an LUR Ž resp. WLUR. point of B if for any x 4
X n B X, the
condition
implies
xn
x
lim 1
n 2
ž n /
lim x x 0 resp. w-lim Ž x x.
0 ;
n
n
Ž. c an almost LUR Ž ALUR.Žresp. weakly almost LUR Ž WALUR..
point of B if for any x 4 B and x 4 B , the condition
implies
X n X m X*
xn
x
lim lim x 1
m n 2
n
mž /
ž n /
lim x x 0 resp. w-lim Ž x x.
0 ;
n
n
Ž d. a midpoint LUR Ž MLUR. Žresp. weakly midpoint LUR
Ž WMLUR.. point of B if lim x x 1 implies
X n n
lim x 0 resp. w-lim x 0 .
n
n
n
ž n /
We say that a Banach space X has one of the above properties if every
point of S has the same property.
X
Remark 3. In our terminology, the result of 5 mentioned above says
that if every x SX
is a rotund point of B X** , then X is -fragmentable.
n

908
BANDYOPADHYAY ET AL.
Clearly, every rotund point of B is an extreme point, indeed an
X
exposed point of B . But the converse is not generally true. For example,
X
no extreme point of B or B is a rotund point of B or B . However, if
l l l l
1 1
every point of SX is an extreme point of B X, then every point of SX
is a
rotund point of B and the space X is rotund.
X
We have the following diagram for all Banach spaces:
LUR ALUR MLUR
WLUR WALUR WMLUR
It is necessary to prove only ALUR MLUR and WALUR WMLUR.
Indeed let x S be an ALUR Ž resp. WALUR.
X
point of BX
and
lim x x 1. Then for every x* S such that x* Ž x.
n n X*
1, we have
lim x* Ž x . 0. So lim x* ŽŽx Ž x x.. 2. 1, hence lim x
n n n n n n 0
Ž resp. w-lim x 0. .
n
In the study of continuity of metric projections, L. P. Vlasov 19 Žsee
16, Theorem 2.
showed that X* is rotund if and only if the union of any
unbounded nested sequence of balls in X is either the whole of X or an
open half-space.
In Theorem 6, we show that this result follows from the properties of
rotund points rather than extreme points, by obtaining a localization of
this property to characterize rotund points of B X *. A variant of this was
obtained in 4, Theorem 2
and seems to have been overlooked since then.
Our proof is simpler.
Later F. Sullivan 17
introduced a stronger property, called Property
Ž V . Žcalled the Vlasov Property in . 2 and showed that a Banach space X
has the Property Ž V . if and only if X* is rotund and X is HahnBanach
smooth. The following reformulation of the definition comes from 2,
Proposition 3.1 .
DEFINITION 4. A Banach space X is said to have the Vlasov Property,
if for every unbounded nested sequence B 4
n of balls and x*, yn S X* ,if
x* is bounded below on B , and the sequence inf y Ž B .4
n n n is bounded
below, or, specifically, if there exists c such that
x* Ž b. c for all b B n ,
then w-lim y x*. n
y Ž b.
c, for all b B , n k
k
In 2 , it is shown that variants of the Vlasov Property characterize the
w*-Asymptotic Norming Properties Ž w*-ANPs .. See 2 for the details.
Here we also show that a weaker version of the Vlasov Property can be
localized to characterize rotund points of B X *.
n

LOCALLY UNIFORM ROTUNDITY 909
In Corollary 8, we show that for any Banach space X, a point x SX
is
a WALUR point of BX
if and only if x is a rotund point in S X** .Itis
known 1 that l has no equivalent WMLUR norm. Hence the norm
12
i 2
ž Ý i/
Ž i.
i1
x x
2 x , x x l ,
is a rotund norm in l which fails to be WALUR. In
6 , it is shown that
every separable Banach space has an equivalent norm with at most
countably many MLUR points and by the construction of the norm, none
of the MLUR points are WALUR points. In Corollary 12 below, we show
that there exists an ALUR Banach space which fails to be WLUR. We
raise the following two questions:
QUESTION 1. Does eery separable Banach space admit an equialent
norm with property , introduced in 12 , without rotund points in the unit
ball?
Let us remark that evidently l1
has property and its unit ball has no
rotund point. In 18
it is proved that if X is weakly compactly generated
then there exists in X an equivalent norm with respect to which X has
property . In 7 this result is extended for Banach spaces with a
fundamental biorthogonal system. In 13
it was shown that any Banach
space with property fails to have LUR points of its unit ball.
QUESTION 2. Does eery WALUR space admit an equialent LUR norm?
A negative answer to this question will give an example of a -fragmentable
Banach space X such that Ž X, weak.
endowed with the metric of the
norm has no countable cover by sets of small local diameter in the sense of
8 . It is known that a WMLUR space with countable cover by sets of small
local diameter is LUR renormable 14 .
DEFINITION 5. Let X be a Banach space. We say that x* SX * is a
w*-ALUR point of B if for any x 4 B and x 4
X * n X* m B X, the condition
x n x*
lim lim Ž xm.
1
m n 2
implies w*-lim x x*. n
ž /
We now present the main results of this paper.
THEOREM 6. Let X be a Banach space. For x* S X *, the following are
equialent:
Ž. a x* is a rotund point of B X *;
Ž b.
x* is a w*-ALUR point of B ;
X *

910
BANDYOPADHYAY ET AL.
Ž. c for eery unbounded nested sequence B 4
n of balls such that x* is
bounded below on B , if for any y 4 S , the sequence inf y Ž B .4
n n X* n n is
bounded below, then w*-lim y n x*;
Ž d. for eery unbounded nested sequence B 4
n of balls such that x* is
bounded below on B n, if y* SX* is also bounded below on B n, then
y* x*;
Ž. e for eery unbounded nested sequence B 4
n of balls such that x* is
bounded below on B , B is an affine half-space determined by x*.
n
n
Ž. Ž. 4 4
Proof. a b . Let x B and x B such that
n X* m X
ž /
x n x*
lim lim Ž xm.
1.
m n 2
As x 4 B , x 4
n X* n has w*-cluster points in B X* . Let y* be a w*-cluster
point of x 4. Then y* 1. It follows that
n
ž /
x* y*
lim Ž xm.
1
m 2
and hence, Ž x* y* . 2 1. By Ž. a , we conclude that y* x*. This
implies the sequence x 4
n has a unique w*-cluster point x*, that is,
w*-lim x n x*.
Ž b. Ž c .. Let B 4
n be an unbounded nested sequence of balls such
that x* is bounded below on B . Let y 4 S , such that inf y Ž B .4
n n X* n n
is bounded below. Suppose c is a common lower bound. Let B n
BŽ x , r .
n n . We may assume without loss of generality that 0 B 1. If we now
put y x r , it follows that y 1. The fact that B 4
n n n n n is nested
implies that y Ž y .
n m 1 crm
for all n m. It follows that
x* y n
ž / Ž ym.
1 crm
2
for all n m. Since r
m
, we conclude
ž /
x* y n
lim lim Ž ym.
1.
m n 2
Hence, w*-lim y n x*.
Ž. c Ž. d . Apply Ž. c to the constant sequence y y* 4
n .
Ž d. Ž e .. Suppose B 4
n is an unbounded nested sequence of balls such
that x* is bounded below on B B . Put inf x* Ž B .. Then
B x : x* x H.
4 Ž .
n

LOCALLY UNIFORM ROTUNDITY 911
If B H, there exist z H and y* S such that inf y* Ž B. y* Ž z.
X *
Ž say .. Thus, y* is also bounded below on B. ByŽ d, . x* y*. But then,
y* Ž z. x* Ž z. inf x* Ž B. inf y* Ž B.
,
a contradiction.
Ž. e Ž. a . Suppose there exists y* S x* 4 such that Ž x* y* .
X *
2
S . Let x 4 B such that Ž x* y* .Ž x . 2. Then, in fact, x* Ž x .
X * n X n n
1 and y* Ž x . 1. Choose a sequence 4
n n such that n 0 for all n
and Ý
n1 n
1. Passing to a subsequence if necessary, we may assume,
x* Ž x . 1 and y* Ž x .
n n n 1 n.
Ž
n n
Let B B Ý x , n Ý .. Clearly B 4
n i1 i i1 i n is an unbounded nested
sequence of balls. For any n ,
ž /
n n n
Ý Ý Ý
inf x* Ž B . x* x n x* Ž x . 1
n i i i i
i1 i1 i1
n
Ý
1 x* Ž x . 2 2.
i1
n
Ý
i i i
i1
Similarly, inf y* Ž B .
n 2. That is, both x* and y* are bounded below
on B B . Now, if B x : x* Ž x. inf x* Ž B .4, then
n
x : x* Ž x. inf x* Ž B. 4 x : y* Ž x. inf y* Ž B . 4.
It follows that ker x* ker y*, and hence, x* y* for some and
since x*, y* S , x* y*.
X *
As an immediate corollary, we get the quoted result of Vlasov with a
much easier proof.
COROLLARY 7 Ž Vlasov ..
X* is strictly conex if and only if the union of
any unbounded nested sequence of balls is either the whole of X or an open
half-space.
Proof. Since X* is strictly convex if and only if every x* SX * is a
rotund point of B , this is immediate from Theorem 6 Ž. a Ž. e.
X *
COROLLARY 8. Let X be a Banach space. For x S X , the following are
equialent:
Ž. a x is a rotund point of B X **;
Ž b.
xisaw*-ALUR point of B X **;
Ž b.
x is a wALUR point of B X ;
Ž. c for eery unbounded nested sequence B 4
n of balls in X* such that
x is bounded below on B , if for any y 4
n n S X** , the sequence
inf y Ž B .4 is bounded below, then w*-lim y** x;
n n n

912
BANDYOPADHYAY ET AL.
Ž c. for eery unbounded nested sequence B 4
n of balls in X* such that
x is bounded below on B , if for any y 4 S , the sequence inf y Ž B
.4
n n X n n
is bounded below, then w-lim yn
x;
Ž d. for eery unbounded nested sequence B 4
n of balls in X* such that
x is bounded below on B n, if any x** SX**
is also bounded below on
B n , then x x**;
Ž. e for eery unbounded nested sequence B 4
n of balls in X* such that
x is bounded below on B , B n n is an affine half-space determined by x.
Proof. The equivalence of Ž. a to Ž. e is Theorem 6, while Ž. b Ž b.
and
Ž c. Ž c. are immediate. And Ž b. Ž c. follows similarly as Ž b. Ž c.
of
Theorem 6.
Ž c. Ž d .. Let B BŽ x , r .4
n n n be an unbounded nested sequence of
balls in X* such that x and some x** S X **, x** x, are both bounded
below on B n , by some c . It follows that for all n 1,
inf xŽ B x n . nŽ x.
rn c
inf x** Ž B x** x n . Ž n.
rn c.
Assume without loss of generality that 0 B . Let x 1 0 SX * be such
that x Ž x x** . Ž say.
0
0. By Goldstein’s Theorem, there exists
x 4 B such that
n
X
and for all n 1,
Then
x Ž x x** . 2
0 n
x Ž x x** . 1.
n
ž / ž /
n
x x n
n 1 c 1
1 x
n Ž xn.
x** 1 .
r r r r
n n n n
Hence, x 1. Putting y x x , it follows that y 4
n n n n n S X. More-
over,
inf x Ž B . x Ž x . r x
n n n n n n
x Ž x** . 1 r x
It follows that
n n n
Ž . Ž .
c 1 r r x c 1 r 1 x
c 1.
n n n n n
c 1
inf ynŽ Bn .
x n

LOCALLY UNIFORM ROTUNDITY 913
and so, inf y Ž B .4 is also bounded below. By Ž c .
n n , w-lim yn x, and
therefore, w-lim xn
x as well. In particular,
lim x Ž x x.
0
n
0 n
and therefore, for sufficiently large n,
x Ž x x** . x Ž x x. x Ž x x** . 2 2 ,
0 0 n 0 n
a contradiction.
LEMMA 9. Let Ž X, 1. be a Banach space and let 2
be a seminorm
on X** such that
Ž. a 1 2
is an equialent norm on X;
Ž b. 2 B is w*-continuous for any bounded subset B in X**.
Then x** x** x** for all x** in X**.
1 2
Proof.
First we show that for every x** X**,
x** x** x** . Ž 1.
1 2
Let x** X**. By Goldstein’s Theorem, there exists a net x 4
in X
such that w*-lim x x** and lim x 1 x** 1 for all . Since
2
is w*-continuous on every bounded subset in X**, we get lim
x
2
x**
. Hence
2
x** lim x lim x lim x x** x** .
Ž. So 1 is proved. Put
1 2 1 2
B x** X** : x** 14,
C x** X** : x** x** 14,
D x X : x 14.
1 2
From the definition of and Ž. 1 , we get D C B. Taking into account
w*
that D B and C is w*-closed, we get B C.
DEFINITION 10. A Banach space X is said to have the KadecKlee
Ž KK.
property if the weak and norm convergent sequences in S coincide.
PROPOSITION 11. Let X be an infinite-dimensional Banach space such
that X* is separable. Then there exists an equialent norm
on X such that
Ž X, . has the KK property and ŽX, . ** is rotund, but ŽX, . fails to be
WLUR.
X

914
BANDYOPADHYAY ET AL.
Proof. Since X is separable, from a theorem of M. Kadec Žsee, e.g., 3,
Proposition 2.1.4, Theorem 2.2.6 . , there exists an equivalent norm on
X with the KK property.
Pick x X and x* X* such that x* Ž x . x x* 1. Let
0 0 0
x
x* Ž x. x x* Ž x.
x , x X.
1 0
We claim that Ž X, 1. also has the KK property. Indeed, let x
n 1
x 1 and w-lim x x. Then lim x* Ž x . x* Ž x ..So
and
1 n n n
Ž .
w-lim x x* Ž x . x x x* Ž x.
x
n
n n 0 0
Ž .
lim x x* Ž x . x lim x x* Ž x . x
x* Ž x.
n
n n 0 n 1 n
1
n
x x* Ž x. x 0 .
Ž .
Since X, has the KK property, we get
Ž . Ž .
lim x x* Ž x . x x x* Ž x.
x 0.
n
n n 0 0
Hence lim x x 0. Thus Ž X, .
n n
1 has the KK property.
Pick a bounded sequence x 4 which spans X*. For x** X**, define
k
12
k
2
2
ž Ý Ž Ž k ..
/
k1
x** 2 x** x
and for x X, set x x1 x 2. Since Ž X, 1.
has the KK property,
we get that Ž X, . also has the KK property. Since 2
is w*-continuous
on bounded subsets in X**, by Lemma 9, for all x** X**,
x** x** x** . Ž 2.
1 2
Now, we show that Ž X, . ** is rotund. Indeed, let x**, y** X** and
x** y**
Ž x** y** . 2 1. This and Ž. 2 imply that
Hence there exists c 0 such that
x** y** x** y** .
2 2 2
x** Ž x . cy** Ž x ., k 1,2,... .
k
k
Since x 4
k spans X*, we get x** cy** which implies that c 1. So
x** y** and hence Ž X, . ** is rotund.

LOCALLY UNIFORM ROTUNDITY 915
Ž .
It remains to show that X, is not WLUR.
Since X is infinite-dimensional for each n , there exists
n
n Ž .
ž k/
n 0
k1
x ker x* ker x , and x x .
From the choice of x , we get w-lim
n n n
x 0. Hence
lim x 0 and lim x x x . Ž 3.
n
n 2 0 n 2 0 2
n
Since x x x x x x , it follows from Ž. 3 that
Since
n n 1 n n 2 0 n 2
lim x x . Ž 4.
n
n 0
x x x x x x
0 n 0 n 1 0 n 2
Ž. Ž.
from 3 and 4 , we get
x* Ž x . x x x* Ž x x . x x x 2
0 0 n 0 n 0 0 n
x* Ž x . x x x 2
,
0 n 0 n
lim x x x* Ž x . x x 2
2x .
Ž .
Hence X, is not WLUR.
n
0 n 0 0 0 0
COROLLARY 12. There exists an ALUR Banach space which fails to be
WLUR.
Proof. Let Ž X, . be the space from Proposition 11 which fails to be
WLUR. By Corollary 8, Ž X, . is WALUR. Since ŽX, . has the KK
property, we get that Ž X, . is ALUR.
ACKNOWLEDGMENTS
This paper was prepared during the visits of the first- and fourth-named authors to the
University of Iowa in the fall semester of 19992000 and the spring semester of 19971998,
respectively. Both of them acknowledge their gratitude for hospitality and facilities provided
by the University of Iowa. Partially supported by Grant MM-80898 of NSF of Bulgaria.

916
BANDYOPADHYAY ET AL.
REFERENCES
1. G. Alexandrov and I. Dimitrov, An equivalent weakly midpoint renormings of the space
l , in ‘‘Proc. 14th Spring Conference of the Union of Bulgarian Mathematicians Sunny
Beach,’’ 1985. In Russian
2. P. Bandyopadhyay and A. K. Roy, Nested sequence of balls, uniqueness of HahnBanach
extensions and the Vlasov property, preprint.
3. R. Deville, G. Godefroy, and V. Zizler, ‘‘Smoothness and Renormings in Banach Spaces,’’
Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 64, Longman,
Brunt Mill, 1993.
4. J. R. Giles, Strong differentiability of the norm and rotundity of the dual, J. Austral.
Math. Soc. Ser. A 26 Ž 1978 ., 302308.
5. J. Giles, P. Kenderov, W. Moors, and S. Sciffer, Generic differentiability of convex
functions on the dual of a Banach space, Pacific J. Math. 172 Ž 1996 ., 413431.
6. B. Godun, Bor-Luh Lin, and S. L. Troyanski, On the strongly extreme points of convex
bodies in separable Banach spaces, Proc. Amer. Math. Soc. 114 Ž 1992 ., 673675.
7. B. Godun and S. L. Troyanski, Renorming Banach spaces with fundamental biorthogonal
system, in Contemporary Mathematics, Vol. 144, pp. 119126, Amer. Math. Soc., Providence,
1993.
8. J. Jayne, I. Namioka, and C. Rogers, -fragmentable Banach spaces, Mathematika 39
Ž 1992 ., 161188, 197215.
9. P. S. Kenderov and W. B. Moors, Fragmentability of Banach spaces, C. R. Acad. Bulgare
Sci. 49 Ž 1996 ., 912.
10. P. S. Kenderov and W. B. Moors, Fragmentability and sigma-fragmentability of Banach
spaces, J. London Math. Soc. 60 Ž 1999 ., 203223.
11. I. Kortezov, ‘‘Fragmentability and Sigma Fragmentability of Topological Spaces,’’ Ph.D.
Thesis, Math. Institute, Bulg. Acad. Sci., Sofia, 1998. In Bulgarian
12. J. Lindenstrauss, On operators which attain their norm, Israel J. Math. 1 Ž 1963 ., 139148.
13. J. P. Moreno, Geometry of Banach spaces with Ž , .-property or Ž , .-property, Rocky
Mountain J. Math. 27 Ž 1997 ., 241256.
14. A. Molto, ´ J. Orihuela, and S. L. Troyanski, Locally uniformly rotund renorming and
fragmentability, Proc. London Math. Soc. Ž 3. 75 Ž 1997 ., 619640.
15. A. Molto, ´ J. Orihuela, S. L. Troyanski, and M. Valdivia, On weakly locally uniformly
rotund Banach spaces, J. Funct. Anal. 163 Ž 1999 ., 252271.
16. E. Oja and M. Poldvere, ˜ On subspaces of Banach spaces where every functional has a
unique norm preserving extension, Studia Math. 117 Ž 1996 ., 289306.
17. F. Sullivan, Geometric properties determined by the higher duals of Banach spaces,
Illinois J. Math. 21 Ž 1977 ., 315331.
18. W. Schachermayer, Norm attaining operators and renormings of Banach spaces, Israel J.
Math. 44 Ž 1983 ., 201212.
19. L. P. Vlasov, Approximative properties of sets in normed linear spaces, Russian Math.
Sureys 28 Ž 1973 ., 166.