25
Internat. J. Math. & Math. Sci.
(1990) 25- 30
VOL. 13 NO.
ON THE STRUCTURE OF SUPPORT POINT SETS
E. AZOFF
Department of Mathematics
University of Georgia
Athens, Georgia 30602
R. YOUNIS
Department of Mathematics
Kuwait University
P.O. Box 5969
13060 Safat
Kuwait
(Received September 7, 1988)
ABSTRACT. Let X be
a metrizable compact convex subset of a
determine the structure of the support point set of
X
when
locally convex space. Using Choquet’s Theorem,
X
has countably many extreme points.
wc
We also
characterize the support points of certain families of analytic functions.
KEY WORDS AND PHRASES: Support point, Extreme point, Choquet’s Theorem.
1980 SUBJECT CLASSIFICATION
1.
(1985 Revision): Primary: 46A55, 52A07, Secondary: 30C99.
INTRODUCTION.
Let X be a subset of a locally convex space E
with fE X if
of X
A continuous linear functional J
ReJ(f)=max{ReJ(g):g E X} and ReJ is
non constant on X.
The set of support points of X will be denoted by Supp X
In this
on
X is said to be associated
case we call
asuormrtpoint
The set of extreme points of a convcx subset F of
E will be denoted by Ext F
Let D
C} and equip the space A of functions analytic
{z: [z[ <
convergence on compact subsets of D
is induced by a sequence
A [1,
{bn}n__ 0
This topology is metrizable
which satisfies lim sup
in D with the topology of uniform
[1, p.1]. Every continuous linear functional
]bnl 1/n <1 and J(0
oo
E
n=0
o0
anbn
for f(z)=
E
on
A
an zn e
n=0
p.36]. Recently, the support points of many subclasses of A have been studied. For more details see [1] and
[2].
Ill Section 2, we consider a metrizable compact convex set X in a locally convex space. Using Choquet’s theorem
we determine the structure of
Supp X when Ext X is countable (Theorem 2.1).
E. AZOFF AND R. YOUNIS
26
In Section 3,
n-1
p
]an] <_ 1},
an zn e A
_<
p
<
n-I
.
In Theorem
Indeed, it is shown that Supp X is in 1-1 correspondence with a proper subset of
3.4, we determine Supp P(p)
Supp
{f(z)
P(p)
we consider the classes:
Ball(p)
SUPPORT POINTS OF SETS WITH COUNTABLY MANY EXTREME POINTS.
2.
Let E be
by Choquct
that
locally convex space, and suppose that X is a metrizable compact convex subset of E
a
[3, p.19]
A theorem
X supported by Ext X such
says that if x E X then there exists a probability measure Px on
I
L dpx for every L in E* In case Ext X is countable (possibly finite), we have the following:
Ext X
CIIOQUET’S THEOREM (Countable Case). Suppose Ext X {fn} is countable. Then X {’ Anfn: A n _> 0
L(x)
!i
An
for each n and
n
PROOF. Let
(/
I
L(f)
such that
1}.
L
X
By Choquet’s Theorem, there exists a probability measure pf
dpf.
[
#f(fn) L(fn) Hence L(fn
Thus L(f)
{fn}
L in E*,
Since’this is true for every
we
get f
n
Pf(fn) fn
npf(fn)fn)
on
X supported by {fn}
O.
as required.
We proceed to the main result of this section.
THEOREM 2.1.
Ext X
{fn
Let
X be
a metrizable
compact convex subset of a locally compact space
(1) Supp X is contained in the union of those
.
Kn
(2)
C_ Supp X if and only if
PROOF. To prove (1), let f
and
<
A
Kn
. .
which are proper subsets of X.
closed affine hull of
fn
(fi:i :/: n}.
Supp X. By Choquet’s Theorem,
n}. Then
{fi:
equal to the closed convex hull of
we can write
Hence we must have Re
Re b(f)
is non-constant on
X,
we must have
Re
A Re
0
t(fi)
Re (f) whenever A > 0. On the other hand, since
b(fi)
Re b(t’) for some i. We conclude that A
b(fi)
>
with each A
Air
Let b be a continuous linear functional associated with f. Then Re b(f)
A Re (f)
Re
Kn
is countable. For each positive integer n, set
E such that
0 for some i, as
required.
To prove (2), suppose that
H g is
_
a closed real
functional J
in
b(fn) =b-
while
fn
subspace of E not continuing
b(fi)
b. This shows that g
Kn
infinite. By. assumption,
i=2
i=2
S
Re b(f)
for every
f/i fi
with
C_ Supp X
fi
:/: n} and fix g
{fi:
Ku
(
Then
A version of the Hahn-Banach theorem [4, page 59] gives
g
.
H-g while b(fn -g)=-1. Set
n. Thus,
fi ->
0 and
SuppX.
b(g) =b for all g
i
1.
Note that S is
Re b(O, we have Re
a
(/
Thus
Kn
b(fn+l)=
a
b. Then
For any h in X.by
b(h)
/n(b-1) +
in
For ease of notation we take n
is a support point of X. Let
{g E E: Re b(g)= Re b(f)}
2il- _1 Re b(fi) < i=2
fn
vanishes on
b(fn+l) =b
Conversely, assume that
set
.
E* whose real part
Choquet’s Theorem, we have h
=b-/n
does not belong to the closed affine hull H of
and assume Ext X
{fn}n=
/i b
is
be an associated linear functional in E* and
closed affine subspace of E.
b(fi)= Re b(f)
Since
Re b(f)=
for alia >2. Thus the closed affine hull
27
STRUCTURE OF SUPPORT POINT SETS
of
=
{fi’i
Thus
1} C_ S. On the other hand, in
S and consequently,
fl
view of Choquet’s Theorem, if
EXAMPLES. (1) Let X be
a
=
{fi:
closed affine hull of
fl
Re
E S then
fl
would l,c constant
o,,
X.
1}
triangle in R2 with vertices
and
fl’ f2
Tl,,,sc vertices are the extreme
f3"
points of X and the affine hull of any two of them is a line, not containing the third. The theorem guarantees that
3
Supp X J Kn which is indeed the boundary of X
n=l
(2) Let X be a square in R2 with vertices
R2
In particular, each
fact, Supp X
fi
/ am,le
hull of
-{f::
:/: i}. The theoren guarantees that
(3) Let T be the fmnily of all functions which
an zn ,an >0.
zrim2
By[5],Ext. T={fn}=l where
of the Theorem. Since
fl
D
are analytic and univalent in
is a limit point of the remaining
{fi}
so
and take the for,u f(z)
for n>l. For ,>l,itis
K n C_ Supp X by the second part
n=2
X and Supp X
K
fi’s
zl
and fn(z)=z-
fl(z)=z
does not belong to the closed affine hull of the remaining
fn
is all of
is contained in S,,pp X.
Kn
no
fi’s
the boundary of X has no interior.
x>
clear that
The affine hull of any three of the
f4
and
fl’ f2’ f3
[.J Kn by the first
part of
n=2
the Theorem.
COROLLARY 2.2. Let X be
as in Theorem 2.1. Then
Supp X
U -6 (Ea)
where each
Ea
Ext X.
PROOF. Suppose
that
(/
Supp X and
is an associated linear functional with f. Writing
whenever A :/:0. Take
Re(f)=Re(fi)
The theorem says these
Ea
Then ffi(Ea) C_ SuppX.
Ea={fi ]A :/:0}.
are proper subsets of
.
is a subset of
Aifi, we see
i.e., they cannot be "too big". The next proposition
Ext X
implies that they can’t all be singletons, i.e., "too small"
PROPOSITION 2.3. Let X be
a compact convex subset of a
locally convex space. If X ha.s more tl,an two
extreme points, then Supp X is uncountable.
X
PROOF. Without loss of generality we may assume that 0
of X, and let
Define
Since
:X
1
and
be continuous and linear functionals such that
2
R2 by ,(f)
(X) has uncountably
(bl(f), 2(f)).
Then
(X)
Let
fl
and
f2
l(fl) 2(f2)
be two independent clenents
and
l(f2) d.2(fl)
0.
is a compact convex subset of R 2 with non empty interior.
many boundary points, Supp((X)) is uncountable. Since
-1
takes support points to
support points, we see that Supp X is uncountable too.
2ri
EXAMPLE. Take
llere all the
Ea’s
fn
e-’n’-
1, 2
and X
-6
{fn}
in R 2
c
Then Supp X
U
n:l
co
{fn,fn+l}
have cardinality two even though Ext X is infinite.
COROLLARY 2.4. Let X be
points.
for n
as in Theorem 2.1. Then
Ext X
Supp X if and only if X has two extreme
E. AZOFF AND R. YOUNIS
28
3.
SUPPORT POINTS OF CERTAIN CLASSES OF ANALYTIC FUNCTIONS.
_<
For
<
p
{E an zn
define P(p)
Co
compact convex subsets of A
A:
E lanl p -<
n=l
n=l
These classes are closely related to
to-onc correspondence with a proper subset of
Ball(gp)
a
are
and we will find that Supp P(p) is in one-
As a corollary, we determine the support l)oints of ccrtain
Supp Ball(p)
{an}x)=l
fanilies of univalent functions. We use the notation a for the sequence
Ve begin with
It is easy to see that the clscs P(p)
1]
simple observation.
is a.sociated with x, then
(x)
X
{x
PROPOSITION 3.1. Let X be the unit ball of a Bausch space E. Then Supp X
llxll
l}. If
I111
PROOF. That every vector of norm one belongs to Supp X is a consequence of the tlahn-Banach theorem.
Suppose conversely that the real part of
X* achieves its maximum over X at x Since X is closed under
multiplication by scalars of absolute value at most one, we have Re
_< IIll Ilxll
and so
IIxll-
(x)
(x)- I111 implies Re (x)_> I(x)l,
1. Moreover
EXAMPLE. The family P(p) "looks like" the unit ball of
p
so
is not a support point of
Re (x)
I111
(x)is in fact real
l}n=l
belongs to the unit sphere of
CO
] an zn
Thus
but we cannot immediately apply Proposition 3.1
n}n=
to filld its support points. For example, the sequence
I111
Re (y)
sup
P(2). The problem is that any non-constant
linear functional
(bn)-I
2
2
bu
which
I1
a.sulne
its maximum at
correspond to
{an}nco=l
must be a scalar multiple of
a continuous linear functional on
{an}n=l.
So lim sup n
bn
which does not
A.
We find the support points of P(p) by making the remarks in the preceeding example more precise.
PROPOSITION 3.2. Suppose T
spaces E and F
E
F is
and let X be a subset of E
a linear, injective, and continuous map between topological vector
Then Tx
Supp TX if and only if x
Supp X and
some linear
functional associated with x belongs to range T*
PROOF. Recall that T* F*
with
E* is defined by
Re(Tx)=max Re(Ty). Set b= ,oT; then
yX
implies that Re
is not constant on
Re (Tx)
max
yTX
(y), and Re
range T* Re b(x)
max
Re (y)
cannot be constant on TX since Re
PROPOSITION 3.3. Let a X-Ball(p),(1 <p<oo),with
(1) If b
is associated with a, then there exists
an lanl p-1
(2) If bn=
{
then b is associated with a.
if
SuppTX and choose
max
yX
F*
Re(y),andinjectivityof T
X
range T* such that Re b(x)
Conversely, let
oT. Suppose Tx
T*
an :/:0
otherwise
f
0 with
T
Write
b is not constant on X.
Ilallp= 1,and
q
/lbnl -lanl
p
b
gq.
for all n.
Then:
F*
Then
STRUCTURE OF SUPPORT POINT SETS
29
(I)-I)(I
It,is r(.stil ft)llovs
froJ llol<h,r’s l.q,ta.lit>..
’Fh( following is the tlai, res,.sl/, of finis sc(:tio}.
cx;
a.z n
TIIEOI,IGNI 3.,I. I.,.l. f(z)
(1)
is a],alyLi(: i,
13
a.,ad
la, P
E
,
is a support l)oitt ot7 1)(i)) if
"I’h.,,
be i P(l))
i1=
<p<
fo
,.
n=l
N
(e) f(z)
M,,,.eo,’(,,"
tr ,,.,,y
<
a.d
’F is
ll61(lcr’s i,cqualil.y,
is tl,, seq,,ent:e
’r-e t)
t-),t,
t.,,,},,__:
if2 :’,l
Clearly T
,
ep,
(1 < p < ),
,-,,ember of
(;o,,sidered a.s
(ii
we h,.v,
sup
IzlSr
(tp)"
lanl o
(:or, vers(dy, spl.,os,:r l.hat,
fllbnl
T(a)
q
a,
I)
or(, (t.r
or (:o,,vergen,:( of tl,,:,
E ra.nge ’["
Thus
I,,,I,
rt’(a)(z)l
{bn},,l
I3
with
(/1).)*
(/p).
then lim
rot all n. Thus lin sup
is analytic in
6_
’l’h,ts
S,l,
"Jla.,,nl <
lanlP
I.
is i,} the
a,,d
’l’h(
<
wc gt,
lly Prop,)sitio,
i ,.,,a.lyti(: i,,
S,Pt)lla.ll (/LI))I.)y
choose th,. hrctional associated with a as in the formul::t of l"rol;osilio,t 3.:. Si(,(, lh(
i,ower
series ot"
i; g/’t,t/.er
than o,,e, lira sup
, I.-
<
so lit,,
Sttpp 1’(i)) 1)y I’rol.),:),-_’itio, 3.2.
I,y Propositions 3.2
=1
Ilbll
finitely many n as rtquir,d.
(7o,,versely, s,,ppose Ta= f(z)=
N
11=1
bn
0
by I.’r,.)l)(,sitio :.1.
,lb,,I
(2) S,,ppos,,
Define
-’’1
n=l
65 Sttpl) Ba.il
n=l
radi,.Is
Ball(ep)onto
Thus {I.,,,} 1,=1
t(l
Supp P(p). By Propositio, 3.2,
0 s,,,:.h i.l,at
l’ropoxit.io 3.1, an(!
,al)s
n=
If the t’,,tctiona.l socited with Ta, is given by
:.3, t.t,,re exists
h)r !)
(;oatinuots. Similarly for p
(1) Suppose !’- .!’
p
an zn
A by T(a)=
la,l
n
t---
t’.ooF,
N
where N is some positive int.eger and
a,,,
ot he.rvise
anZ n a,,d
N
11::]
I,,!
t. Then
e S,pp I:-.II.
E. AZOFF AND R. YOUNIS
30
COltOLI, ARY 3.5.
(1)
is aualytic in D and
(2) f(z)
PROOF.
+
N
+
A function f(z)
hlanl
al,
p
n is a support point of
<p<
if
,
zn
Q(p) if and only if
,,
2, if p
for om I,ositive integer N
I.1
an and
n=2
n=2
n
One way to see this, is to replace {p by ep(H) where it(n)
n
in the proof of
2,3
Theorem 3.4.
R.EMARK. One
can define
P(o)
an zn suplanl <
{f(z)
!i--"
to tl,e proof of Theorem 3.4, that.
Supp P(oo)
{f(z)
:
,:X:3
n=l
anz n lanl
One can show. usi,,s
a,,
arg,,n,c,,t
si,,,ilar
for some n > 1}.
REFERENCES
1.
SCIIOBER, G. Univalent Functions Selected Topics, Springer-Verlag, 1975.
2.
IlALLENBECK, D.J. and MACGREGOR, T.H. Linear Problems and Convexity Techniques in Geometric Function
Theory, Pitman, 1984.
3.
I’ItELPS, R.R. Lectures on Choquct’s Theorem, Van Nostrand, 1966.
4.
R,
UIIN, W.
Functional Analysis, McGraw-llill, 1973.
5. SILVERMAN, H. Univalent functions with negative coefficients, Proc. Amer. Math. Soc. 5t(1975), 109-116.
6.
SILVERMAN, tl. Order starlikeness for multipliers of univalent functions, J. Math. Anal. Appl. 103(1984) 48-57.