Advances in Pure Mathematics, 2011, 1, 30-32
doi:10.4236/apm.2011.12008 Published Online March 2011 (http://www.SciRP.org/journal/apm)
Copyright © 2011 SciRes. APM
Relative Widths of Some Sets of m
p
l*
Weiwei Xiao, Weijun Luan
College of Sciences, North China University of Technology, Beijing, China
E-mail: wwsunny@163.com
Received January 14, 2011; revised January 30, 2011; accepted March 10, 2011
Abstract
In this paper, the relative widths of some sets in m
p
l are studied. Relative widths is the further development
of Kolmogorov widths and it is a new problem in approximation theory which aroused some mathematics
workers great interest recently. We present some basic propositions of relative widths and investigate relative
widths of some sets (ball or ellipsoid) of m
p
l.
Keywords: Kolmogorov Widths, Relative Widths
1. Introduction
In 1984, V. N. Konovalov in [1] first proposed the
definition of relative widths which is in the sense of
Kolmogorov. Let W and V be centrally symmetric
sets in a Banach space
X
. The Kolmogorov n-dimen-
sional widths of W relative to V in
X
(shortly,
relative widths) is

,, :=,
sup
inf inf
nX
nn
fW
LgVL
KWVXf g

where the infimum is taken over all n-dimensional
subspaces n
L of
X
, nN. When =VX the rela-
tive widths coincid es with the n-dimensional Kolmogo-
rov widths (shortly, nK widths) of W in
X
, which
we denote by

,
n
dWX. Of course,

,, ,
nn
K
WV XdW X
for any set V, and if 12
VV, then

12
,,,, .
nn
K
WV XKWVX
Y. N. Subbotin and S. A. Telyakovskii in [7-9], V. M.
Tikhomirov in [11], V. F. Babenko in [2-4], V. N. Kono-
valov in [1,5,6], V. T. Shevaldin in [10] etc. gained many
results in this field. And some Chinese mathematics
workers such as Yongping Liu, Lianhong Yang in [15-17]
and Weiwei Xiao in [12-14] also did some work on rela-
tive widths.
Let m
p
l, 1p, denote space of vect o rs

1
=,,
m
x
xx with norm
1
1
=, 1<
p
pp
m
pxx p
x
1
=max,,, =.
m
xxp
x
Let
:=: 1
m
pp
p
Bl
xx be the unit ball in m
p
l.
Let
1
=,,
m
diag DDD be an mm real diagonal
matrix. Without loss of generality we assume that
12 >0
m
DD D. Let
M
be a positive real num-
ber, set
=:, ,
m
pp
M
RMDx xx
obviously it is ellipsoid in m
p
l. When =1
M
, we denote
it by
p
.
Theorem A: [19] For 1p
, 1<mn,
1
,= ,
m
npp n
dlD
Similar to the proof in [18] we can get the following
proposition.
Proposition 1.
1) If W is a finite set of
m
elements, then for the
linear spanning subspace
lin W one has
 

,,= ,,=0
nn
KW lin WXKlin Wlin WX
for nm.
2) If 1
WW, then
 
1,,,, .
nn
K
WVXK WVX
3) For any scalar
, and any W and V, one has

,,= ,,.
nn
K
WVX KWVX
 
4)
 
012
,,,,,, .KWVXK WVXKWVX
5) Let 0
=m
WK
, where 0
K
is a bounded set and
*Supported partly by National Natural Science Foundation of China
(No.10471010) and partly by the project “Representation Theory and
Related Topics” of the “985 Program” of Beijing Normal University
and partly by Beijing Natural Science Foundation (1062004).
W. W. XIAO ET AL.
Copyright © 2011 SciRes. APM
31
m
is a subspace of dimension m. If <nm, then

,, =
n
KWVX .
6) For the convex hull
co W, if for each subspace
n
X
of dimension n,

n
co WX is a locally sequen-
tially compact and closed subset, then


 
,,= ,,.
nn
K
WcoW XKcoWcoW X
7) If Y is a subspace of
X
and WYX ,
VY, then

,,,, .
nn
K
WV XKWVY
Theorem 1 For >mnN, 1p
, 11
>n
DD
,
the smallest number
M
which makes the equalities

1
,,= ,=,
mm
np pp nppn
KMldlD
  (1)
hold is 1
01
:= 1n
D
MD
, and


10
10
1,0<<,
,,= ,.
m
np ppn
M
DMM
KMl DMM


Theorem 2 For all mN such that >1m,

111 1
,, =.
2
m
m
KBBl

2. Proof of Theorems
Proof of Theorem 1: For

01
=,0,,0Dx, we have

0
1
,, sup
inf inf
sup inf
inf
=1.
m
np pp
p
nm n
x
LlyM L
p
pp
p
yM
xp
p
p
yM p
KMl
DM




 xy
xy
xy
That is

1
,,1, 0<1.
m
np pp
KMlDM M  (2)
In order to make the equalities (1) hold, we have that

11
1,
n
DDM

that is 1
1
1n
D
MD
 . (3)
For 1
1
0<1n
D
MD
 , we will prove that

1
,, 1.
m
np pp
K
Ml MD (4)
For each =
p
xDz, 1
pz, set
1
=,,,0,,0 .
n
np
yMx MxLM When =p
,
the inequality (4) is trivial, so we only need to prove the
case of 1<p
.








 
11
111 1
11
1111
11
1
=1
1
11
11.
pp ppp
p
nn m
p
p
pp
pp
nnn nmm
pp
ppn
pp
pp
ppp
nn m
ppp
pp
p
Mxx xx
M
DzDzDzD z
MDzz
MDMDDzz
MDz MD


 

 
 
 


xy
In fact, when 1
1
0<1n
D
MD
 , we have

11
11
11
n
MDD
. So we get that inequality (4).
By inequalities (2) and (4), we have that


11
1
0<1, ,,=1.
m
nnp pp
D
M
KMl MD
D
  (5)
From (3) and (5) we get that the smallest number
M
which makes the equalities (1) hold is 1
01
=1 n
D
MD
.
For 0
M
M,
 
01
,,, ,=.
mm
n pppn pppn
KMlKMlD
  (6)
By Theorem A, for all >0M,
 
1
,, ,=.
mm
nppp nppn
KMldlD
  (7)
From (6) and (7) we get
10
,,=, .
m
np ppn
K
Ml DMM

The proof of Theorem 1 is complete.
Proof of Theorem 2: From [6] we know that

111 1
,, .
2
m
m
KBBl

(8)
We want to prove that

111 1
,, .
2
m
m
KBBl

(9)
W. W. XIAO ET AL.
Copyright © 2011 SciRes. APM
32
By proposition (6) we know that

1111 1
,,=,, ,
mm
mm
K
BBlK WBl

(10)
where

=0 ,,0,1,0,,0:=1,,, represent the th coordinate
i
Wimii .
Set

112
:=:= 0.
mm m
LRxxx
x
For

=0,,0,1,0, ,0
i
aW, set

11
1
=0, ,0, 12,12,0, ,0,
m
ii
bLB

=1, ,id, when =id, 1i represent the 1st coor-
dinate, we get =12ab
. So we proved

11
,, 12,
m
m
KWBl

which means that inequality (9) is valid. The proof of
Theorem 2 is complete.
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