Séminaire Choquet.
Initiation à l’analyse
I VAN S INGER
Some open problems on best approximation in normed linear spaces
Séminaire Choquet. Initiation à l’analyse, tome 6, no 2 (1966-1967), exp. no 12, p. 1-8
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Séminaire CROQUET
(initiation à 1~ Analyse)
6e année, 1966/67, n° 12
12-01
16 .,février 1967
SOME OPEN PROBLEMS ON BEST APPROXIMATION IN NORMED LINEAR SPACES
by Ivan
[according
to
a
SINGER
SINGER]
by Ivan
draft
Notations.
E : normed linear space
G : linear subset of
E ~
x E
E
go
(real
complex).
or
E.
G : best approximation of
llx - g0~
0
nearest
or
.
G
is
best
if
particular,
In
1. possess
is
Does
segment
no
unit cell of
E
x E E
linear
a
has
For
proof
of the
m =
3 ,
E
eby0161ev
G
subspace
possess
polyhedron
is
a
a
(e.
G
line
(x
E
(dim
a
cube) ,
exist
a
only
one)
1) ?
such that there
answer
-
A. L.
=
the
problem. - Student ~ KLEE himself
McMINN [15]
dim G
parallell to
is
1964,
and
1,~
=
G ? If the
is
obviously afagain obviously affirmative.
KAKUTANI, CHOQUET
and others.
has
proved that the answer is affirmative. Another
given by A. S. BESICOVITCH [1].
incorrect proofs have been given by G. EWALD
ROGERS and others.
In
one
subspace.
through the origin
E ~I
g. take
m > 3 ,
2. -
e.
Cebysev
E = n t
answer
T. J.
(i.
following probl em (in somewhat difBanach space E, where 3 $ m oo ~
m-dimensional
of this result has been
For
one
has raised the
firmative. For the usual euclidean norm, the
History
exactly
G
subspace,
the unit sphere
on
is
a
KLEE [13J
one-dimensional
a
,
G
G .
E
Does every
Geometrically :
exists
G
V.
1957,
terms) :
ferent
if each
subset,
approximation g0
In
))x -. g!)
inf
point.
Cebysev
a
=
if
x
GARKAVI [10]
has
[7],
G.
ROGERS,
EWALD-
posed the following problem : Does there
separable Banach space which has no Cebysev subspace ?
*
Since,
at least
in every
Banach space
one
jjxj)
separable conjugate
exposed point ( x ~ E ~
H
hyperplane
of
S
to any
!!xjj
among those
E
seek for the space
(x e E )I
=
E
space
Cebysev subspace :
no
=
E(l)
non-complete
space of
have
myself
)!x()
norm
given
a
a
x
n
S~.
support
a
M )y
==
one
must
not isometric
are
of scalars which have at
endowed with the usual
E
no
~-
the
no
vec-
Cebysev subspace.
following example of
Cebysev
sequences (i.
which has
I > c , then the
cardinality
has
sup
iel
=
E
Banach space
non-separable
{03BEi}i~I
=
of all almost-zero
finite number of
set of
(unpublished)
normed linear space
c.. consisting
0 , except
is
a
"coordinates"
non-zero
and with the
operations
I
If
of all bounded families
countable number of
a
V. KLEE and
=
there exists
has
S-
conjugate space.
which has
tor
the unit cell
Banach spaces which
separable
GARKAVI has constructed the following example of
most
B ,
=
1 , such that
satisfying H
=
l)
~
E
separable
a
subspace : the
e.
dense sub-
whose coordinates
are
them).
v
E
It is well known that the space
has
Co
no
eby0161ev subspaces
l])
==
subspaces of any finite dimension. It is also known that
no eby0161ev subspace of finite dimension or of finite codimenCebysev subspaces. Thus it is natural to ask the following
has
sion, but yet it has
problem : How to combine the spaces c.. and L ([0 y l])
parable Banach space which has no eby0161ev subspace ? [.The
are
not
not
isometric,
3. -
A related
eby0161ev subspace
question of GARKAVI : Does
(codim
It is well known that it has
4. plex space
This
a
codimension
spaces
conjugate Banach
the space
c.. and
space. ]
C([0 y 1])
a se-
L1([0 , 1])
possess
(of
1 ,
G
=
dim
a
~
Cebysev subspaces
but has
no
problem presents
the classical theorem of
of any finite
Cebysev subspaces
those
compact
all continuous functions
Cebysev subspace
E/G) ?
v
(topologically)
Characterize
C(Q)
in order to obtain
of infinite dimension and infinite codimension
v
subspaces of
n ~ 2 .
to any
isomorphic even,
.
tains
of infinite di-
but it has such
mension,
E
==
on
Q
dimension,
and
eby0161ev
of finite codimension
spaces
Q
with
jjx
for which the
=
max
)x(q)j )
comcon°°°°°°~
q~Q
of finite dimension
some
interest from the
Banach-Stone,
n
> 1 .
following point of view : Although
shows, theoretically, that the vectorial-metric properties of the spaces C(Q) are
completely determined by the topological properties of the compact spaces Q and
conversely yet the effective, explicit study of this interdependence still presents
many open problems, and a solution of the above problem might be considered as a
contribution to this study.
According
dimensional linear
eby0161ev
a
(complex) theorem,
Kolmogorov
an
n-
subspace
G
is
(real) -
to the classical Haar
=
and
subspace if,
C(Q)
... ,
only if,
x y
... ,
is
x
Cebysev system
a
Q
on
n
(i.
each linear
e.
combination ~
cy.
has at most
x.
1
n -
zeros
Q ).
on
i=l ~ ~
the above
In the
which admit
Q
compact spaces
of real scalars
case
following : Characterize
to the
problem amounts
Cebysev system
a
(i.
(topologically)
S. MAZUR has
e.
with
x y ... , xn
Thus
those
n >
1 .
that if a
conjectured
v
,
compact space
Q
is
admits
Q
to
homeomorphic
a
Cebysev system
subset of
by J. C. ?.IAIRHUBIIIR
first time
subset of
a
real
a
a
[l6]
x
circle. This
under the additional
k-dimensional euclidean space. For general
case
of
1 , then
proved for the
n >
has been
that
hypothesis
Q
compact spaces, it
P. C. CURTIS
T. YANG
with
x
conjecture
proved, independently, by K. SIEKLUCKI [20] and
proof has been given by I. J. SCHOMBERG and C.
For the
... ,
y
[4-]~
A
more
is
a
has been
simple
[l9j.
complex scalars, only the following partial result of I. J.
(lac. cit.)
SCHOENBERG and C. T. YANG
is known : If
compact
a
Q
space
admits
a
~
complex Cebysev system
a
subset of
dean
a
finite
with
... ~ x
polyhedron,
then
Q
n >
is
1 ,
and if
Q
homeomorphic to
a
is
homeomorphic to
subset of the eucli-
plane.
Other
problems.
1° Similar
GARKAVI
[10]
problem
has
2° Extension to
~. 2014
for
codimension
Concerning
[or
n ~
this
L1(T ,
v)
L-(T .
spaces. For real
03BD) ,
A. L.
proved...
dim
Characterize
space
complex
the
k
(x
e
E ~ G) .
(topologically) those compact
complex space C(Q) ] contains
spaces
a
Q
Cebysev
for which the real
subspace of finite
1 .
problem,
there exist only
partial results
of R. R. PHELPS
[18 ]
GARKAVI ([8J, [9J
and A. L.
conditions
1 . Let
~
Q 9
on
us
giving sufficient conditions
in order that
C (Q)
contain
Cebysev
a
or
necessary
subspace of codimension
recall them :
1° Sufficient conditions.
(a)
Q
(b)
Q
set of
compact separable
(in particular, compact metric) implies
compact separable, satisfying
its isolated points), implies
Q
=
(i.
Q ’ Q’
(n ~ 1, ,
codim G ~ n
The
ble)
compact spaces
Q
at the International
announced that if
Q
is also
2°
n ~
Necessary
1 .
Thus,
Mathematicians [1966. Moscow],
metric, then the condition of (b)
contain
counta-
Q
However,
A. L. GARKAVI has
is both necessary
Cebysev
subspaces of any finite
compact spaces, the problem is solved.
for metric
co-
conditions.
(Cebysev)
(c)
ble number of
...) .
not necessary.
are
of
and sufficient in order that
dimension
2 ,
(in particular, for
subsets (continua).
showing that the above conditions
Congress
the closure of the
=
this condition
satisfying
may also contain connected closed infinite
There exist examples
Q
e.
.
disjoint
codim G
imply
(imply,
open subsets
in
that
Q has at most
particul ar,
at most
a
a
counta-
countable
num-
points).
ber of isolated
(d)
and
(Cebysev)
G
and
codim G =
n >
1
that
imply
Q
contains
no
open
an
infinity of
con-
connected infinite subset.
(e) Corollary
nected
(d) :
Q
is
finite,
G
Characterize those Banach spaces E
with
codim
The set of all best
or,
when either
or
Q
has
components.
6. space
of
equivalently :
with the
approximation
of
which contain
no
following property :
x,
such that there does not exist
closed linear sub-
For
R. C. JAMES
1y
n =
C12~
(the proof
has shown
difficult) that this
[his first proof for se-
is very
class coincides with the class of reflexive Banach spaces
diable
spaces [ 11 ] was not correct, a stndent
error in it]. I have posed this problem
tional
Congress
parable
G: each
set
It is obvious that in
a
C ebys ev
set. I. S.
reflexive
a
E
x
MOTZKIN [17]
f
E
E*
withjjfj)
tion with this
result,
=
case
of
to Professor JAMES at the Interna-
exactly
e.
for every
eby0161ev
every
convex
Cebysev
set is
and
n
has shown that for
convex.
in
a
fi-
there exists
exactly
In
connec-
set is
convex.
oo . N. V. EFIMOV and
n
at the
Colloquium
Convexity
on
of the above is also
converse
Cebysev set is
However, A. BRNDSTED
finite-dimensional Banach space in which every
there exists
n >, 3 ,
n =
a
set is
convex.
n-dimensional Banach spaces,9 where
must be smooth. For
([2], [3])
set is
that
e.
E ~9
E
x
go E G.
convex
proved that,9 inversely,
STECKIN [5] and V. KLEE ( ~ 1.4 ~ or a Conference
[1965. Copenhagen] to appear) have claimed that the
i.
approximation
Banach space, every
convex
and others have
(i.
best
one
S. B.
true,
irreme-
following problem arises naturally : Characterize those
Banach spaces in which every
1° The
has
f(x) = ~x~ ),
1 ,
the
E
strictly
nite-dimensional smooth Banach space
one
an
0
Moscow, ~966.
of Mathematicians in
7. - Cebysev
of V. KLEE has discovered
a
=
n >, 3
non-smooth
BR~NDSTED (~3 ~
this is
2 ,
obviously true.
this is
false ;9 namely,
for every
integer
n-dimensional Banach space in which every
has also solved the above characterization
3 , by proving that every C ebys ev set in
only if, every exposed point of SE = fx
( with
E
3)
dim E =
E E i~~x!
1~
is
a
is
Cebysev
problem for
convex
smooth
if,
point
of
SE .
For
n >
2° The
3 ,
case
is known.
the
problem
’
is still open.
of infinite-dimensional Banach spaces. In this case,
Not only
the
possibility
sional smooth Banach spaces is
of
extending MOTZKIN’s
unknown,
but
the
even
considerably
less
result to infinite-dimen-
answer
to the
following ques-
space ~ , is every Cebysev set necessarily convex
The best result known until the present is the following theorem of L. P. VLASOV
~21 ~9 which contains as particular cases results of V. KLEE, N. V. EFIMOV and
tion is unknown : In
a
Hilbert
?
v
S. B. STECKIN : In
a
smooth Banach space,y every
ded closed subset of it is
and
elegant,
it makes
use
conpact) Cebysev
and S. B. STECKIN
[6J
set is
of the Schauder fixed
Since the condition of being boundedly
boundedly compact
convex.
The
(i.
proof
e.
every boun-
is very short
point theorem.
compact,
is too
have introduced the notion of
restrictive, N. V. EFIMOV
approximative compactness : a
is called
G c E
set
approximatively compact if ,
for every
E
x E
and every
se-
quence
p(x ’ G)
with lim ~x-gn~ =
there exists
a
{gnk}
subsequence
Banach space
E,
formly
convex
pact).
N. V. EFIMOV and S. B.
convex
Banach space
E.,
Banach space, the
vex
(1)
(2)
(3)
every
(by
above),
the
is
and
is
convex
G
is
G
is
Cebysevian
Cebysevian
convex
for
that in
proved
have
following statements
G
(e.
G
element of
an
g. in
are
G
set
a
and
if,
in
a
Cebysev
of
only if, it is approsmooth
uniformly
con-
equivalent :
and
weakly closed ;
and
approximatively compact.
which
sets in smooth
show,
uniformely
among
convex
others,
following problems :
(a)
(b)
Is
a
eby0161ev
set
weakly closed ?
Is
a
eby0161ev
set
approximatively compact ?
imposing supplementary
conditions
directly
order to be able to conclude that it is convex,
to each
that the
Banach spaces is
the
(assigning
com-
closed ;
eby0161ev sets,
Instead of
uni-
smooth uniformly
a
We have thus useful infinite-dimensional characterizations of closed
in terms of
a
weakly closed subset is approximatively
STECKIN [6]
Cebysev set G
a
ximatively compact. Hence
converging to
,
one
can
on
convex
sets
problem of convexity
to each of
equivalent
the
eby0161ev
set
impose conditions
G
on
in
the
best
approximation of x in G ;9
this is well defined, once G is a eby0161ev set). Thus, V. KLEE has proved [14]
that, if G is a Cebysev set in a smooth and reflexive Banach space E y and if
admits a neighbourhood V(x) such that
is both contiG is convex. The condition of continuity of the
nuous and weakly continuousy then
where G is a eby0161ev set, does not imply the weak continuity of
mapping
as shown by the example
mapping
the
x
unique
s
v
~.
E
on
the other
is
a
The
the
(some
=
Hilbert
space) ,
G =
SE ;;
hand, it is not known whether the weak continuity
eby0161ev set, implies its strong continuity.
problem of convexity of
following problem
property
that every
maximal distance from
on
Cebysev
farthest
points :
admits
does
sets in Hilbert
a
A
If
a
closed
of
set
unique f arthest point in
necessarily consist
of
one
G
is also related to
spaces ?
convex
where
A
A c z
( i.
e.
has the
which has
simple point ?
A. F. FICKEN and V. KLEE
have shown that if the
C 1~ ~
y
Since the
pose the
problem
(Conference
dered two such
has at most
a
G
convexity
Cebysev
of
at the
Colloquium
C2 . Let
C1-set if G
a
in
approximation
smooth Banach space
E
is
question would
is open, it is natural to
some
larger classes of sets.
Convexity [1965. Copenhagen])
and
C1
to this
convex.
sets
belonging to
A
on
answer
would be
set in ?
convexity of sets
is called
best
one
G)
is
and
easily
C1 : A
mention
us
set
G ,
has consi-
in
a
normed
semi-Cebysevian (i. e. each x
closed. Every boundedly compact
shown to be
However,
convex.
C1-
V. KLEE has
space H , there exists a C
non-void, bounded and convex (hence
complementary set H B G is
whose
E
E
in every infinite dimensional Hilbert
shown, that,
set
of
classes,
E ,
linear space
set in
of
problem
V. KLEE
Cebysev
then every
affirmative,
be
~,
G
non-convex).
is
Some other
interesting related problems
have been raised in the
same
paper of
V. KLEE.
REFERENCES
(A. S.). - On the set of directions of linear segments on a convex
surface, Convexity [1961, Seattle], p. 24-25. - Providence, American mathematical Society, 1963 (Proceedings of Symposia in pure Mathematics, 7).
[2] BRØNDSTED (Arne). - Convex sets and Chebyshev sets, I, Math. Scand., København,
[1]
BESICOVITCH
t.
17, 1965,
p.
5-16.
[3] BRØNDSTED (Arne). t.
[4]
[5]
Convex sets and
p. 5-15.
18, 1966,
Chebyshev sets, II,
Math.
Scand.,
København,
CURTIS
J. of
(Philip C., Jr). -
EFIMOV
(N. V.) and STE010CKIN (S. B.). - Support properties of sets in Banach spa010Ceby0161ev sets [in Russian], Doklady Akad. Nauk SSSR, t. 127, 1959, p.
t.
Math.,
9, 1959,
n-parameter
p.
families and best
approximation,
Pacific
1013-1027.
and
ces
254-257.
(N. V.) and STE010CKIN (S. B.). - Approximate compactness and 010Ceby0161ev
[in Russian], Doklady Akad. Nauk SSSR, t. 140, 1961, p. 522-524.
[6]
EFIMOV
[7]
EWALD
(Günter). - Über
Hamburg,
[8]
GARKAVI
nal
(A. L.). p.
of
die Schattengrengen konvexer
p. 167-170.
Körper,
Abh. math. Semin.
On best approximation by the elements of infinite-dimensioa certain class [in Russian], Mat.
Sbornik, N. S., t. 62,
104-120.
GARKAVI (A. L.). - Approximation properties of subspaces of finite defect in
the space of continuous functions [in Russian], Doklady Akad. Nauk SSSR, t.
155, 1964,
[10]
27, 1964,
subspaces
1963,
[9]
t.
sets
p.
513-516.
GARKAVI (A. L.). - On
Akad. Nauk SSSR, t.
010Ceby0161ev
and almost
28, 1964,
010Ceby0161ev
p. 799-818.
subspaces
[in Russian],
Izv.
12-08
[11]
(R. C.). - Reflexivity
JAMES
Math.,
[12]
Series
(R. C . ) . -
JAMES
23, 1963,
[13]
[14]
KLEE
2,
p.
66,
t.
and supremum of linear
1957, p. 159-169.
Characterizations of
reflexivity,
functionals, Annals of
Studia
Math., Warszawa,
t.
205-216.
(Victor L.). - Convex sets, Bull. Amer. math. Soc., t. 63, 1957, p. 419.
(Victor L.). - Convexity of Chebyshev sets, Math. Annalen, t. 142, 1961,
KLEE
p. 292-304.
[15]
NcMINN (Trevor J.). - On the line segments of
fic J. of Math., t. 10, 1960, p. 943-946.
[16]
MAIRHUBER
a convex
surface in
(John C.). -
problems having
615.
On Haar’s theorem concerning Chebyshev
unique solutions, Proc. Amer. math. Soc., t.
E3 ,
Paci-
approximation
7, 1956, p. 609-
[17]
MOTZKIN (I. S.). - Sur quelques propriétés caractéristiques des ensembles convexes, Atti Accad. naz. Lincei, Rend., Série 6, t. 21, 1935, p. 562-567.
[18]
PHELPS (R. R.). J. of Math., t.
[19]
SCHOENBERG (I. J.) and YANG (C. T.). - On the unicity of solutions of
of best approximation, Annali di Mat. pura e appl., Serie 4, t. 64,
1-12.
[20]
SIEKLUCKI
systems,
(K.). -
010Ceby0161ev
subspaces of finite
13, 1963, p. 647-655.
codimension in
C(X) ,
Pacific
problems
1961,
p.
Topological properties of sets admitting the Tschebycheff
Sc., Sér. math. astron. et phys., t. 6, 1958,
p.
Bull. Acad. polon.
603-606.
[21]
VLASOV
Nauk
(L. P.). SSSR,
t.
010Ceby0161ev
141,
sets in Banach spaces
1961, p. 19-20.
[in Russian], Doklady
Akad.