Internat. J. Math. & Mh. Si.
533
(1978)533-539
Vol.
DENTABLE FUNCTIONS AND
RADIALLY UNIFORM QUASI-CONVEXITY
CARL G. LOONEY
Department of Mathematics
University of Toledo
Toledo, Ohio 43606
U.S.A.
(Recieved April 18, 1978)
ABSTRACT. In this paper we give a further result which states sufficient conditions for the theory of convergence of minimizing sequences to be applicable,
develop the theory further, and give an application.
KEY WORDS AND PHRASES. Deting points, de,table function, radial quasiconvexity, normed spaces, convex domains, minzing sequences.
AMS (MOS) SUBJECT CLASSIFICATION (1970) CODES.
Primary
46B99.
1.
49D455 Secondary 52A05,
INTRODUCTION.
Let X be a normed space, C a closed bounded convex set in X, and f: C+R
A minimizing sequence for f
a (nonlinear) functional to be minimized on C.
on C is a sequence
(Xn)
n)
in C such that f(x
/
8
inf f(C).
In [4] and [6], certain conditions were given which guarantee that any
534
C. G. LOONEY
minimizing sequence of f on C will converge in norm to a minimum (when it exists).
In this paper we give a further result which states sufficient conditions for the
theory of [4] to be applicable, develop the theory further, and give an application.
SOME DEFINITIONS AND THEOREMS
2.
The closed convex hull of a set S in X is denoted by cl-conv(S).
is dentable at x e S whenever: given any e >
B (x) is the open e-ball centered on x.
0, x
We say S
(SB (x)), where
cl-conv
In this case x is called a denting
[2; p. 97]).
point of S [5] (strongly extremal point of S,
See [9] for the
origin of the term "dentable".
A function f on a convex set C is said to be dentable [4] at x e C iff
0
(x0,
f(x0))
{(x,)
is a denting point of epi(f)
X
-->
R
f(x)}.
It
was shown in [4] that if f is a 1.s.c. (lower semi-continuous) quasi-convex
C,
functional on a weakly compact convex set C and has a unique minimum x
0
then every minimizing sequence of f converges in norm to x iff f is dentable
0
at x
0.
=<
f(x)
{x e C:
We say that f is quasi-convex on C iff the level sets L
are convex.
f(%x + (i -%)y)
<=
max
This is equivalent to the following:
{f(x), f(y)}, 0
=<
for any x, y e C,
Convex functionals are quasi-
% < i.
convex, but not conversely.
A normed space X, its closed unit ball, and its norm are all said to be
lxll
uniformly convex iff given e > 0 and x, y with
II x Yll
> e, there exists
()
space is strictly convex, i.e.,
i.e.,
l%x +
(I
p
It is shown [3] that the and L
0 < % < I.
for
> 0 such that
< p <
.
(x,)
P
0, then we say that
II
i,
lYll -<
l1/2x + Yll -< I
%)Yll < %11xl + (I
I and
().
Every such
%)IIYlI,
spaces are uniformly convex
depends on the point x also,
When the modulus of convexity
>
=<
is
locally uniformly convex [8]
Locally uniformly convex spaces are not generally uniformly convex, but the
converse is true.
Also, locally uniformly convex spaces are strictly convex,
DENTABLE FUNCTIONS AND QUASI-CONVEXITY
535
but not conversely.
THEROEM i.
point of B
I(0)
PROOF.
is a denting point of B
Let x have norm i.
Qg, flY
Y e
6.
i
If X is a locally uniformly convex space, then every boundary
xll
>
Qe
BI(0)B
(x,e)
so that there is some
BI_ (0)
The set
I(0).
Given e > 0, let
For any
(x).
I11/2
> 0 such that
I being closed and containing
(Q)HI
cl-conv
LEMMA I.
Qe.
H
2
cl-conv
e C.
Then
(Qs).
Let C be a compact convex subset and f: C
0
Yll
I and H 2
contains x as an interior point.
closed convex set, so that x
a unique minimum x
+ 1/2
can be strictly separated from x by a closed hyper-
plane (see, e.g., [3; p.193]) H which partitions X into two halfspaces H
with H
x
R be l.s.c, and have
Then for any e > 0, f is bounded away from
f(x0)
B
on
CB g (X ).
O
PROOF.
on Q.
The set Q
By hypothesis, y > B.
CONSTRUCTION.
(Xo)
CB
is compact and thus f attains its
Thus f(x)
->
y >
B
Let f assume its infimum
infSmum y
on Q.
at a unique point x
0
e C
closed
bounded convex subset of a normed space X that is locally uniformly convex.
{(x,B): x
set H
{(x0,):
line
e
e X} meets epi(f) at the point
R} in X
a distance r above (x
on
Pr"
Put E
r
Then
Br(Pr
{(x,):
THEOREM 2.
0
x
).
R.
(x0,B).
Fix r and take the point
Let B
r
(Pr)
(x0
Pr
+ r)
e L at
(x0,8).
8+r}epi(f).
Let f be a l.s.c, functional on a convex bounded set C and let
0
e C.
If X is a locally uniformly convex normed space,
then a sufficient condition for f to be dentable at x
0
ii)
Let L be the vertical
be the closed ball of radius r centered
meets H at the unique point
f have a unique minimum x
i)
there exist some r > 0 such that
Er be
is that either,
contained in
Br(Pr )’
or
C be compact.
PROOF.
i)
The
Since (x
0
f(x0)
is a denting point of B
r
(Pr)
in X
R (by
536
C. G. LOONEY
I(x
Theorem 1 and the fact that the product norm
remains locally uniformly
convex) and
Er Br (pr)
e)
II
(I Ixl 12x +
it is clear that
leI2) 1/2
(x0,
cl-conv(E{--Be(x0, f(x0)))Ccl-conv(Br(Pr)Be(x0, f(x0))) for any e
ii) For any e > 0, f is bounded away from f(x0) on CBe(x0).
f(x0) is seprated by a closed hyperplane from epi(f)(Be(x0, f(x0)
srong,
The conditions of Theorem 2 part i) are
f (x
0)
> 0.
Thus
and as the next example
shows, may not be satisfied even in a finite dimensional space.
-I/x 2 for
e
EXAMPLE i. Let f(x)
0. Then f is a conx # 0 and f(0)
tinuous quasi-convex function R, but we take C to be the compact set
The point x
in R
2.
r
0 is the unique minimum.
Let
Br(Pr
The bottom branch of the sphere of B
r
(pr)
We show now that llm f(x)/g(x)
x
It can be shown that f’ (0)
nbhd of 0.
0
Pr
is a convex function
0, so that f(x)
<
f"(0), by putting
t
using the limit definition for derivatives at x
f" (x) / g" (x)
be centered on
[-I, I].
0.
(0,r)
g(x)
g(x) on some
I/x and
Then lim f(x)/g(x)
0/(r 2) -1/2 (by use of L ’Hospital’s Rule twice).
r, there is some e > 0 such that epi(f) is not contained in B
lim
-
Thus for any given
r (pr)
for
< x <
Although f does not satisfy the hypothesis of Theorem 2, it is dentable by part
ii of Theorem 2.
We note that Theorem 2i can be phrased in terms of the
of f and g, where g(x)
]f’(x;y)
3.
=>
]g’(x;y)
inf
at
x
{e:(x,e)
for all
NEW TYPE OF QUASI-CONVEXITY
r(Pr)},
y
Gteaux
derivatives
i.e., we need
C.
AN IMPORTANT APPLICATION.
We say that a subset S of a vector space is radially convex at s
O
any line L through s
o
meets S in a convex set
LS. Any
S iff
convex set C is
radially convex at any point of C.
A functional f defined on S is said to be, respectively, radially convex,
radially, strictly co.n.vex, radially uniformly convex, or radially locally uni-
DENTABLE FUNCTIONS AND QUASI-CONVEXITY
537
S, whenever S is radially convex at s O and f is, reap.,
convex, strictly convex, uniformly convex, or locally uniformly convex on all
forml
convex at s
segments
LS
o
through s
0.
We replace "convex" by "quasi-convex" to get the
resulting four new definitions for f at s
in the radially convex set S.
o
We now consider the space C[e,8] of all continuous functions on the interval
Is,8]
and the subset of rational functions.
that the approximation functional T (a,b)
p
m+l
n+l
convex when p
where a e R
R
b
a
xm)/(b 0 +
+
m
b
n
xn).
The norm
II’II
It is known(see [I] or [7])
II f(’)
r
(a
mn
and r
(a, b; x)
b;-)
II p
is quasi-
(a0 +
+
is not strictly convex since its
graph on the unit ball contains horizontal llne segments.
THEOREM 3.
lgll
The functional g +
iS radially uniformly quasl-convex
at 0 on any convex nbhd U of 0 in any normed space.
PROOF
II’II
Let L be any line through 0 and put L
is convex on L
I11/2 x + 1/2 Yll
> ]I Yll is the
+ 1/2 y
max
max
0.
{llxll, IIYlI}"
llxll
convexity on L
1/211Yll + 1/211Yll
>
IIYlIllxll
+ 1/2
This completes the proof, since L
0
such that
<
Iixll
IlYll
IlYll
<
I11/2
x
and
IlYll
+ 1/2 YlI,
another
is compact and strict quasi-
0
implies uniform quasi-convexity on L
(see [4; Lemma i]).
0
We note that a norm may not be radially strictly convex, e.g.,
II-II 1
IlYl] (l]xll
In this latter case 1/2 x
Otherwise
1/2
0
e0, eitherllxll
Since x, y
0, which yields a contradiction.
contradiction.
We know that
Now suppose that there are x # y in L
same case) or else
{llxll, IIYlI}
LU.
0
II’II and
are linear on line segments from 0 to any x(x must be in the positive
orthant in the case of
II IIi )-
For 1 < p <
l’llp
is uniformly convex and
therefore is radially strictly convex on any radially convex set w.r.t, to a
point.
THEOREM 4.
The function a +
lf
formly quasi-convex at the minimum a
nbhd of a for f fixed in C [e,8],
(a 0 +
(a
< p <
+
0,...,am)
a
xm) ll p
m
is radially uni-
on any closed bounded
538
C. G. LOONEY
function a
0.
Without loss of generality, we assume that f
PROOF.
/
(a +...+
0
convex with minimum on
amxm)
Rm+l
followed by the convex
The linear
If" II
map
is trivially
at 0. Since the space of polynomials P
m
m+l
degree m or less is linearly isomorphic to R
a
lla0
/
+...+
a
[e,8] of
xmll p
m
is
and thus Theorem 2 holds (it is known that a unique
actually a norm on
minimum a does indeed exist for 1 < p).
We now borrow a proposition from [7].
For f fixed in C [e,8], the functional
PROPOSITION.
(a, b,
b*)
")II
where r
mn
(a0 +...+ a
(a,b x)
T
COROLLARY 2.
is a
0
xm)/(b 0 +..+
T,
> 0 such that T
(a*,
xn).
mizing sequence
re+n+2.
decreases.
0, b 0)
# (a*, b*) there
Further, given a step size of
will decrease by at least
0 along dO
eO
is a
there
until the
,
T
is dentable at its unique minimum
n,
b n)
(a
(a*,
in R
directional minimum is reached.
COROLLARY 2.
r
n
is radially uniformly quasi-convex at the minimum
Starting from any point (a
in which
b
m
b*) of any closed nbhd U
o
lf-
T(a,b)
is quasl-convex on any convex nbhd U of the unique minimum
COROLLARY i.
direction d
See [I] also.
,
converges to (a,
b
(a, b), and any mini-
DENTABLE FUNCTIONS AND QUASI-CONVEXITY
I.
Barrodale, I. Best Rational Approximation amd Strict Quasi-Convexity,
SIAM J. Numerical Anal., I0 (1973) 8-12.
2.
Choquet, G.
3.
K’the, G.
4.
Looney, C. G. Convergence of Minimizing Sequences, J. Mth. Anal. &
Appl.,61 (1977) 835-840.
5.
Looney, C. G. A Krein-Milman Type Theorem fQr Certain Unboumded Convex
Sets, J. Math. Analysis & Appl., 48 (1974) 284-293.
6.
Looney, C. G. Locally Uniformly Quasi-Convex ProgramminK, SlAM J. Appl.
Math. 28 (1975) 881-884.
7.
Looney, C. G.
8.
Lovaglia, A. R.
(1965) 225-238.
9.
Rieffel, M. A.
Lectures on Anaysi_s, oi. II, nJamin, N.Y., 1969.
Topological Vector Slaes I, pringer-Verlag, Berlln, 1969
(Transl. of 1966 German Ed. ).
Rational Approximation and Computation (preprint).
Locally Uniformly Convex Banach Spaces, Trans. AMS
Dentable subsets Qf Banach spces with application to
Radon-Nikodym theorem, Proc. Conf. on Functional Analysis, U. C. Irvime,
Thompson, Washington, D. C., 1967.
539