Banach-Mazur distance
and
$\mathrm{B}$-convex
Banach spaces
岡山県立大学情報工学部
高橋泰嗣
(Yasuji
Tk 市 ashi)
Department
of System Engineering,
Okayama Prefectural University
九州工業大学工学部
加藤幹雄
(Mikio Kato)
Department
of
Mathematics,
Kyushu
Institute
of
Technology
Abstract.
A Banach
space
$X$
is
said
to be
$\mathrm{B}$-convex
if
it
is
$B_{n}$
-convex
for
some
$n\geq 2$
.
As
is well-known,
$\mathrm{B}$-convexity
is
an
isomorphic invariant,
but
$B_{n}$
-convexity is
not
so.
In this short
note,
we are
concerned with the
stability
of
$\mathrm{B}_{n}$-convexity under
norm
perturvations.
It
is known (cf.[7])
that
$X$
is
$B_{n}$
-convex
$(n\geq 2)$
if and
only
if
the n-th
von
Neumann-Jordan
constant
$C_{NJ}^{(n)}(X)$
is
less than
$n$
.
We show that for
isomorphic
Banach
spaces
$X$
and
$\mathrm{Y}$it
holds
$C_{NJ}^{(n)}(\mathrm{Y})\leq C_{NJ}^{(n)}(X)d(X, \mathrm{Y})^{2}$
,
where
$d(X, \mathrm{Y})$
denotes the
Banach-Mazur distance between
$X$
and
$\mathrm{Y}$; and this implies that
if
$X$
is
$B_{n^{-}}$
convex,
then there exists
$\mathrm{A}_{n}>1$
such that all Banach spaces
$\mathrm{Y}$satisfying
$d(X, \mathrm{Y})<\lambda_{n}$
are
$\mathrm{B}_{n}$-convex.
In the
case
$X=l_{p}$
or
$L_{\mathrm{p}}[0,1],$
$1<p<\infty$
,
it
is
also shown
that
all
Banach spaces
$\mathrm{Y}$satisfying
$d(X, \mathrm{Y})<n^{1/f}$
are
$B_{n^{-}}$
convex, where
$r= \max\{p,p^{j}\}$
and
$1/p+1/p’=1$
.
Moreover,
if
$X=l_{\mathrm{p}}^{n}$
or
$L_{\mathrm{p}}[0,1],$
$1<p\leq 2$
,
then
there exists
a
Banach space
$Y$
with
$d(X, \mathrm{Y})=n^{1/p’}$
such
that
$\mathrm{Y}$is
not
$B_{n}$
-convex.
同型なバナッハ空間
$X,$
$Y$
に対し
,
Banach-Mazur distance
$d(X, Y)$
は
$X$
と
$\mathrm{Y}$の近さ
を表すと考えられる.
x,
Y
力 ssOmetric
であれば X のもつ幾何学的性質
(狭義凸性,
$-$
様凸性等)
はすべて
Y
に遺伝する
.
X, Y
がお
Ome
緬
c のとき
d(X, Y)=1
であるが
,
一般
にその逆は成立しない
. d(X, Y)=l
のとき,
狭義厭性は遺伝するとは限らないが,
–
様凸性等の超性質はすべて遺伝する
.
バナッハ空間論では局所的性質,
とりわけ超性質
(super
property) の研究が重要である.
一様凸性, 一様平滑性,
uniform
non-squareness,
tyPc
$\mathrm{p}$,
cotype
$\mathrm{q},$$B_{n}$
-convexity,
$J_{n}$
-convexity,
超回帰性などバナッハ空間の重要な性質の
多くは超性質である
.
無限次元バナッハ空間に関する自明でない任意の超性質を
P
とす
るとき
,
無限次元ヒルベルト空間と
isometric
な空間は性質
P
を有し,
また
,
性質
P
を
:
8
$\mathfrak{l}\mathrm{h}\ovalbox{\tt\small REJECT} 5\Phi\emptyset \text{超}\#\mathrm{H}^{-}\mathrm{C}\hslash O$,
$F\beta \mathrm{E}\sigma$)
cotype
$k\mathrm{b}^{\vee}\supset\sim\vee\geq[]\mathrm{h}\mathrm{E}5\Xi\emptyset \text{超}\#\mathrm{F}^{-}\mathrm{C}\text{あ}6$.
:
:
$\mathrm{V}\ovalbox{\tt\small REJECT}$ト
$;x\mathrm{f}\mathrm{f}\mathrm{P}_{\mathfrak{k}},7\not\supset \mathrm{S}\not\subset \mathrm{f}6$:
$X,Yl\grave{\grave{>}}_{\grave{\mathrm{J}}}\mathrm{E}\mathrm{V}^{\backslash }(d(X, Y)\emptyset\grave{\grave{:}}\text{ノ}\downarrow\backslash \mathrm{g}_{\mathrm{V}^{\backslash }})\text{と}\mathrm{g},$$X\emptyset \text{
超性
}\mathrm{F}[]\mathrm{h}Y[]’\backslash \not\in\ulcorner ET6T$
$\text{あ}6\check{\mathit{0}}\hslash>$
?
$\urcorner \mathrm{p}\#$;
ヒ
,’^‘‘’’
ト
$p_{\mathrm{D}}arrow \mathrm{F}\ovalbox{\tt\small REJECT} l_{2}$es,
$T\wedge^{\backslash \vee}\mathrm{C}\backslash \mathit{0}$)
$\text{超}\uparrow\not\subset \mathrm{f}\mathrm{l}\epsilon\# T6$.
$Y\emptyset^{\theta}\backslash$$l_{2}$
&\Pi
$\ovalbox{\tt\small REJECT} Th\mathcal{X}\iota$VX
$1\leq d(l_{2}, Y)<\infty Tb6$
.
$d(l_{2}, Y)=1f_{\mathrm{e}}\mathrm{C}\mathrm{b}$
If.
$\mathrm{g}\Re_{\backslash },$$Y[]\mathrm{h}\mathrm{f}\wedge^{\backslash }T\backslash \emptyset \text{
超}\#\mathrm{E}k\mathrm{F}T6$
.
$T1\mathrm{h},$
$d(l_{2}, Y)<\lambda ktx6T\wedge^{\backslash }T\backslash \emptyset \mathrm{Y}\delta^{\mathrm{P}}>\text{超性}\mathrm{E}Pk\mathrm{b}’\supset\ddagger\dot{\mathrm{p}}f_{X\lambda}>11\mathrm{h}\#\# 9^{-}6\mathrm{T}$
k6
$\check{\mathit{0}}\hslash>?\text{
超
}$
-convexity
(Z)
\ddagger
$\dot{\mathcal{D}}^{f}X4\mathrm{I}\mathrm{f}\mathrm{f}\mathrm{l}\mathrm{f}9\text{性}\mathrm{F}\}^{-\prime}.\supset \mathrm{t}\backslash \tau|\mathrm{h},$ $\mathrm{g}$as,
$\hslash\not\in\cdot \mathrm{t}6$(A
$>1\dagger\mathrm{h}\mathrm{I}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{k}‘ \mathrm{V}$\ddagger
$\mathrm{V}^{\backslash }$)
.
$1,\hslash \mathrm{l}\mathrm{b}fp\mathfrak{p}_{\backslash \mathrm{b}}^{\theta},$$-\ovalbox{\tt\small REJECT}\ \# b6\mathrm{t}\backslash []\mathrm{h}-\ovalbox{\tt\small REJECT}*\#\#\emptyset$
\ddagger
$\check{\mathcal{D}}’l\mathrm{a}\mathrm{n}l^{\prime^{*-}}\neq \mathrm{f}\mathrm{f}\mathrm{i}\text{性}$$\mathrm{F}[]’.’\supset \mathrm{V}^{\backslash }T1\mathrm{h}\yen \mathrm{f}\mathrm{f}\mathrm{l}l^{\mathrm{i}}\xi ft6$
.
$\not\equiv\ovalbox{\tt\small REJECT}$, Hlk(7)
$\lambda>1[]’.X\backslash \}\mathrm{L},$
$d(l_{2}, Y)<\lambda k^{;}x6\mathrm{Y}T-\mathrm{f}\mathrm{f}\mathrm{i}$
th
$(k6\mathrm{v}\backslash [] \mathrm{Z}-\ovalbox{\tt\small REJECT}^{\backslash \prime}\mp \mathrm{f}\mathrm{f}\mathrm{i})Ttx\mathrm{v}\backslash \mathrm{t}\emptyset\theta^{\theta}\backslash h6$.
$k_{arrow}^{\vee}6T,$
$-\ovalbox{\tt\small REJECT} \mathrm{f}\mathrm{l}\#(\text{あ}6\mathrm{V}^{\backslash }\}\mathrm{h}-\mathrm{f}\mathrm{f}\mathrm{i}*\mathrm{f}\mathrm{f}\mathrm{l}\#)\text{と}$$\text{
超
^{
ロ
}}$
Ea
$r\mathrm{x}\mathrm{f}\mathrm{f}\mathrm{l}_{-\backslash }^{*_{\backslash }},k1_{-J^{-}}\mathrm{C}$uniform non-squareness
$(B_{2^{-}}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{x}\mathrm{i}\mathrm{t}\mathrm{y}$$\text{あ}6\mathrm{V}^{\backslash }$
I:
$J_{2}$
-convexity
$k6\mathrm{p}\mathrm{E}$)
$l^{*}>\text{
あ
}6$
.
$(\text{超}\mathrm{P}\mathrm{m}\mathrm{f}\hslash\#\mathfrak{h}f_{\mathrm{X}\text{空間}\}\mathrm{h}},$ $-\mathrm{E}\iota \mathrm{h}\text{空}\mathrm{P}\mathrm{S}l^{\mathrm{i}}\epsilon-;-\epsilon\tau\sim\tau\emptyset l\mathrm{X}\mathrm{f}\mathrm{f}\mathrm{l}$$\mathfrak{X}\mathrm{i}*\mathrm{E}*\mathfrak{X}\Leftrightarrow T6^{\vee}-kl^{\mathrm{i}}\pi \mathrm{b}\hslash T\mathrm{V}^{\backslash }6$
(Enflo
[1])
$)$.
$\not\in_{\grave{\mathrm{J}}}\mathrm{E}$,
uniformly
non-square
$- \mathrm{e}\text{あ}\epsilon$ $\ddagger\dot{\mathcal{D}}’I\text{空間}\mathrm{f}\mathrm{h}*11\mathrm{A}\mathrm{R}\mathrm{I}\mathrm{t}\mathrm{h}$(fixed point property)
2
$\mathrm{b}’\supset-\sim\not\geq l\dot{\backslash }/\overline{\tau\backslash }\mathrm{S}h$,
$\not\in f_{-}^{-},$$d(l_{2}, \mathrm{Y})<\lambda$
$\mathrm{V}\text{
あ}6$
\ddagger
$\check{\mathcal{D}}’\mathrm{A}T\wedge^{\backslash }T\backslash \emptyset Yl^{\mathrm{i}}T\backslash \mathfrak{U}k\text{性}\xi \mathrm{b}’\supset\ddagger\dot{9}’I5\mathrm{f}\mathrm{l}\emptyset\lambda k_{)}\Re\#\mathrm{S}*\iota T\mathrm{V}^{\backslash }6(\mathrm{c}\mathrm{f}.[2],$$[8]$
,
[9]
$)$.
$k_{\sim}^{-}6T,$
$d(l_{2}, \mathrm{Y})<\lambda Tk6\ddagger\dot{9}’\mathit{1}\tau\wedge^{*}T\emptyset Yl\dot{\backslash }$
uniformly
non-square
$kfX6$
A
$\emptyset \mathrm{E}\lambda \mathrm{m}\mathrm{a}$Sk
$\lambda=\sqrt{2}\mathrm{T}!$ $\text{あ}6(\mathrm{c}\mathrm{f}.[11])$
.
$/\mathrm{J}^{\rangle}\mathrm{f}\mathrm{f}\mathrm{i}\emptyset \mathrm{B}\mathrm{f}\mathrm{f}\mathrm{i}\}\mathrm{h}$
,
uniform
non-squareness
1)
6
V
$\backslash []\mathrm{h}$\ddagger
$\mathfrak{h}-\Re\emptyset B_{n}$
-convexity
$[]’.’\supset \mathrm{v}\backslash \tau$,
$*\sigma)\text{性}\mathrm{f}\mathrm{f}\emptyset\backslash \not\in\not\in \mathrm{f}\mathrm{f}\mathrm{l}\$
Banach-Mazur
eene
$\text{と}\sigma$)
$\ovalbox{\tt\small REJECT} \mathrm{r}\mathrm{f}\backslash *\mathrm{e}\Rightarrow\not\in T6-arrow k,$ $\ovalbox{\tt\small REJECT} \mathrm{t}’.$,
$\mathrm{B}_{n}$-convex
‘i3
$\text{あ}6$
\ddagger
$\dot{\mathrm{p}}tx\mathrm{F}\mathrm{f}\mathrm{f}\mathrm{f}\mathrm{f}\mathrm{i}\prime_{I_{\mathrm{R}}^{*\text{間_{}X\dagger’}}}$.
;kt
$\mathrm{b},$$d(X, \mathrm{Y})<\lambda_{n}\mathrm{C}\text{
あ
}6\mathrm{f}\wedge T\emptyset Y\theta\dot{\backslash }B_{n}$
-convex
$\text{と}t_{X}$$6$
\ddagger
5
$fx\mathrm{B}\mathrm{B}(\mathrm{B}\lambda)\emptyset\lambda_{n}\xi\Re\not\in T6^{\vee}\sim kT\text{あ}6$
.
1. Definitions
(i)
For
isomorphic
Banach spaces
$X$
and
$\mathrm{Y}$,
the
Banach-Mazur
dis-tance
between
$X$
and
$\mathrm{Y}$, dcnoted
by
$d(X, \mathrm{Y})$
,
is
defined
to be the
infimum
of
$||T||||T^{-1}||$
taken
over
all
bicontinuous
linear operators
$T$
of
$X$
onto
$Y$
.
(ii)
A Banach space
$\mathrm{Y}$is
called
finitely representable
$(f.r.)$
in
a
Banach
space
$X$
if for
any
finite
dimensional
subspace
$F$
of
$\mathrm{Y}$and
for
any
$\epsilon>0$
there exists
a
finite
dimensional
subspace
$E$
of
$X$
with
$\dim E=\dim F$
such
that
$d(E, F)<1+\epsilon$
.
(i\"u)
Let
$P$
be
a
property
for
Banach
spaces. We
say
$X$
has super
$P$
if any
Banach
space
$Yf.r$
.
in
$X$
has
P.
$P$
is
called
super property if $P=superP$
.
Of
course,
$X$
is
super-reflexive if any Banach space
$Yf.r$
.
in
$X$
is
reflexive.
2. Definitions
(i)
$X$
is
called
uniformly
non-square (James, 1964) if
there
exists
$\delta>0$
such
that
$\min(||x+y||, ||x-y||)\leq 2(1-\delta)$
if
$||x||=||y||=1$
.
(ii)
The James
constant of
$X$
is
defined
by
It
is
obvious
that
$X$
is uniformly
non-square if
and
only if
$J(X)<2$
.
(iii)
The
von
Neumann-Jordan constant
of
$X$
is defined
by
$C_{NJ}(X)= \sup\{\frac{||x+y||^{2}+||x-y||^{2}}{2(||x||^{2}+||y||^{2})}$
:
$x,$
$y\in X$
, not both
$\mathrm{z}\mathrm{c}\mathrm{r}\mathrm{o}\}$.
It is known that
$X$
is uniformly
non-square
if and only if
$C_{NJ}(X)<2(\mathrm{c}\mathrm{f}.[5],[10])$
.
3.
$B$
-convexity and
$B_{n}$
-convexity
$X$
is
said
to
be
$B_{\mathfrak{n}}$-convex
(or
uniformly
$non-\ell_{1}^{n})$
provided
there
exists
$\epsilon(0<\epsilon<1)$
such
that for all
$x_{1},$
$\ldots,$
$x_{n}\in B_{X}$
there
exist
9
$(\epsilon_{j}=\pm 1)$
satisfying
$||\epsilon_{1}x_{1}+\ldots+\epsilon_{n}x_{n}||\leq n(1-\epsilon)$
,
where
$B_{X}$
denotes the closed
unit
ball
of X.
$X$
is
called
$B$
-convex
if
$X$
is
$B_{n}$
-convex
for
some
$n\geq 2$
.
It
is well-known
that
$X$
is
$B$
-convex
if
and
only
if
$l_{1}$is not
finitely
representable in
$X$
; and
if
and only if
$X$
is
of
type
$p$
for
some
$p>1$
.
4.
Theorem Let
$1<p<2$
.
Suppose that there exists
$\epsilon(0<\epsilon<1)$
such that
for
all
$x_{1},$
$\ldots,$
