数理解析研究所講究録 1455 巻 2005 年 65-69

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Functions related to some geometrical propertiesof Banach Spaces
高橋 泰嗣 (Yasuji Takahashi)

岡山県立大情報工加藤 幹雄 (Mikio Kato)

九州工大工as a generalization (orAbstract. We introduce some functionsrefinement) of the Sch\"affer constant $S(X)$ of Banach spaces $X$ , and investigate geometrical properties of Banach spaces such as uniform nonsquareness and uniform convexity in terms of those functions. The normal.
structure coefiicient $N(X)$ is also estimated by the function$\varphi_{X}(\tau)$$\varphi_{X}(\tau)$

バナッハ空間 $X$ の幾何学的性質の度合いを記述しようとすれば, 幾何学的定数 (あるいは関数) の考察が必要となる. James 定数 $J(X)$ と Sch\"affer 定数 $S(X)$ は uniformは一様凸性の度合いをnon-squareness の度合いを表し, modulus of convexity表す. CJarkson [4] が導入した von Neum nann-Jordan 定数 $C_{NJ}(X)$ は, ヒルベルト$\delta_{X}(\epsilon)$

空間の特徴づけに関連した概念であるが, 最近では $J(X)$ との関係も考察されている (cf.[12]). また, $C_{NJ}(X)$ を用いて uniform non-squareness の特徴づけもできるを用いて一様凸性など(cf.[16]). しかしながら, これらの定数 $J(X),$ $S(X),$
不動点定理に関連した重要な概念
他方,の幾何学的性質を記述することはできない.
である一様正規構造は正規構造係数で記述される. バナッハ空間 $X$ が一様正規構造をもつための十分条件が, James 定数 $J(X)$ や von Neumann-Jordan 定数 $C_{NJ}(X)$ との関連で知られている (cf.[7,12,14]). 本講演では, Sch\"affer 定数 $S(X)$ の概念を一般を導入し, uniform convexity, uniform non-squareness
化 (精密化) した関数による評価等を考察する. 以などの特徴づけ, 更には, 正規構造係数 $N(X)$ の$C_{NJ}(X)\backslash$$\varphi x(\tau)$$\varphi_{X}(\tau)$

下 $X$ をバナッハ空閤とする.

1. Definitions (i) $X$ is called uniformly
there exists $\delta>0$ such that$\prime^{-}/\iota on$-square in the sense of James when$\min(||x+y||)||x-y||)\leq 2(1-\delta)$

if $||x||=||y||=1$ .

(ii) The James constant is defined by$J(X):= \sup\{\min(||x+y||, ||x-y||) :||x||=||y||=1\}$ .

(iii) $X$ is called uniformly non-square in the sense of Sch\"affer when there exists
such that$\lambda>1$$\max(||x+y||, ||x-y||)\geq\lambda$

if $||x||=||y||=1$ .

BB(iv) The Sch\"affer constant is defined by$S(X):= \inf\{\max(||X+y||, ||x -y||) :||x||=||y||=1\}$ .

(v) The von Neumann-Jordan (NJ-) constant of a Banach space $X$ is the smallest
constant $C$ for which$\frac{1}{C}\leq\frac{||x\cdot+y||^{2}+||x-y||^{2}}{2(||x||^{2}+||y||^{2})}\leq C$for$\forall(x, y)\neq(0,0)$

holds; we denote it by $C_{NJ}(X)$ .
(vi) The modulus of convexity of $X$ is defined by$\delta_{X}(\epsilon)=\inf\{1-||\frac{x+y}{2}||$$X$:$||x||=||y||=1,$ $||x-y||=\epsilon\}$is called uniformly convex if
if there is $C>0$ such that$\delta_{X}(\epsilon)>0$$(2\leq q<\infty)$

for all$0<\epsilon\leq 2_{j}$$\delta_{X}(\epsilon)\geq C\epsilon^{q}$

for all$(0\leq\epsilon\leq 2)$.

and -unifromly convex$q$$0<\epsilon\leq 2$.

It is obvious that $X$ is uniformly non-square in the sense of James, resp., Sch\"affer
if and only if $J(X)<2$ , resp., $S(X)>1$ . Since $J(X)S(X)$ $=2$ for any Banach space$X$ (cf.[3,12]), these two notions are equivalent.
It is known that $X$ is uniformly nonsquare if and only if $C_{NJ}(X)<2$ (cf.[16]). Let us recall that $X$ is super-reflexive
if any Banach space finitely representable in $X$ is reflexive. It is well-known that if$X$ is uniformly convex, or moregenerally, uniformly non-square, then $X$ is reflexive.
It is easy to see that if $X$ is uniformly non-square, then any Banach space finitelyrepresentable in $X$ is uniformly non-square. Thus, any uniformly non-square Banach
space is super-reflexive (cf.[9]). Enflo [5] showed that $X$ is super-reflexive if and only
if $X$ admits an equivalent uniformly convex norm. Pisier [13] also showed that if $X$
is super-reflexive, then $X$ admits an equivalent -uniformly convex norm for some$\mathrm{q}$$2\leq q<\infty$.

2. Definitions (i) A Banach space $X$ is said to have normal structure if $r(K)<$diam(K) for every non-singleton closed bounded convex subset $K$ of $X$ , wherediam(K) $.– \sup\{||x-y|| : x, y\in K\}$ and $r(K):= \inf\{\sup\{||x-y|| : y\in K\} : x\in K\}$ .
(ii) The normal structure coefficient of $X$ (Bynum [2]) is the number:$N(X)= \inf${$\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(K)/r\cdot(K)$: $K\subset X$ bounded and convex, diam(K) $>0$ }.

Obviously, $1\leq N(X)\leq 2$ . The space $X$ is said to have uniforrn normal structure
if $N(X)>1.$ It is well-known that if $X$ has uniform normal structure, then $X$ has
fixed point property (cf.[8]). Gao and Lau [7] showed that if $J(X)<3/2$ , then $X$ has
uniform normal structure. Prus [14] even estimated the normal structure coefficient
by $J(X)$ . Kato, Maligranda and Takahashi [12] also estimated the normal$N\acute{(}X)$

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structure coefficient $N(X)$ by $C_{NJ}(X)$ , and showed that if $C_{NJ}(X)<5/4$ , then $X$
as welJ as its dual $X’$ have the uniform normal structure.
3. Definitions ($Sch\dot{a}ffer$type constants): We define for$\tau\underline{>}0$$S_{X,p}(\tau)$$\inf\{(\frac{||x+\tau y||^{p}+||x-\tau y||^{p}}{2})^{1/p}$$:=$:

if $1<p<\infty$ ,$||x||=||y||=1\}$$\{$$\inf\{\max(||x+\tau y||, ||x-\tau y||)$:if $p=\infty$ .

$||x||=||y||=1\}$

Let $X$ be a Banach space (of dimension at least 2). Let be a strictly convex and(such a function
with values instrictly increasing function defined on$\varphi(1)=1$.
For simplicity, we assume that $\varphi(0)=0,$
is continuous on$\varphi$$[0, \infty)$$[0, \infty)$$[0, \infty))$$.$

4. Definition (Generalized Sch\"affer type constant): For$\varphi_{X}(\tau)=\inf\{\frac{\varphi(||\bm{x}+\tau y||)+\varphi(||x-\tau y||)}{2}$$1<p<\infty,$ then
5. Remark If
is the Schiffer type constant.
$\varphi(t)=t^{p},$6. Proposition

7. Theorem$\varphi x(\tau)$$X$:$\tau\geq 0$

let$||x||=||y||=1\}$$\varphi^{-1}(\varphi_{X}(\tau))=S_{X,p}(\tau),$

where$S_{X,p}(\tau)$

is continuous and non-decreasing for $0\leq\tau<\infty$ .

is uniformly non-square if and only if$\varphi_{X}(\tau)>1$

for some$0<\tau<1$ .

8. Corollary Let $1<p\leq\infty$ . The following are equivalent.
(1) $X$ is uniformly non-square.
(2) $S_{X,p}(1)>1$ .
(3) $S_{X,p}(\tau)>1$ $(0<\exists\tau<1)$ .
(4) $S_{X,p}(\tau)>\tau$ $(1<\exists\tau<\infty)$ .

9. Theorem$X$is uniformly convex if and only if$\varphi_{X}(\tau)>1$

10. Corollary Let $1<p\leq\infty$ . The following are equivalent.
(1) $X$ is uniformly convex.
(2) $S_{X,p}(\tau)>1$ $(0<\forall\tau<1)$ .
(3) $S_{X,p}(\tau)>\tau$$(1<\forall\tau<\infty)$.

for any $0<\tau<1$ .

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11. Theorem $N(X)\geq\varphi^{-1}(\varphi_{X}(1/2))$ . In particular, if $\varphi_{X}(1/2)>1$ , then $X$
has uniform normal structure.

12. Corollary Let $1<p\leq\infty$ . Then
$N(X)\geq S_{X,p}(1/2)$ .

It is easy to see that if $C_{NJ}(X)<5/4$ , then $S_{X,2}(1/2)>1$ . Since $C_{NJ}(X)=$$C_{NJ}(X’)$ , we have
13. Corollary If $C_{NJ}(X)<5/4$ , then $X$ as well as $X’$ have the uniform normalstructure.

14. Theorem Let $1<p\leq\infty$ and $2\leq q<\infty$ . The following are equivalent.
(1) $X$ is -uniformly convex.
(2) There is $C>0$ such that$q$

$S_{X,p}(\tau)\geq(1+C\tau^{q})^{1/q}$for all$\tau\geq 0$.

15. Theorem Let $2\leq p<\infty$ . Then the following are equivalent.
(1) $X$ is isometric to a Hilbert space.
(2) $S_{X,p}(\tau)=(1+\tau^{2})^{1/2}$ for all.
$\tau\geq 0$

16. Remark If $X$ is a Hilbert space, then for all$S_{X,p}( \tau)=(\frac{|1+\tau|^{p}+|1-\tau|^{p}}{2})1/p$$L_{r}$$\tau\geq 0$

if $1<p<2$ .

Hence, the above theorem is false if $1<p<2$ . Finaily we calculate-spaces.

17. Theorem Let be an(1) Let $1<r\leq 2$ and$\mathrm{X}$$L_{r}$-space with $\dim X\geq 2$ .
. Then for all$1/r[perp]_{\mathrm{I}}1/r’=1$$\tau\geq 0$$S_{X_{\mathrm{J}}p}( \tau)=(\frac{|1+\tau|^{r}+|1-\tau|^{r}}{2})1/r$

if $r\leq p\leq\infty$ .

(2) Let $2\leq r<\infty$ and $1/r+1/r’=1$ . Then for all$S_{X,p}(\tau)=(1+\tau^{r})^{1/r}$$\tau\geq 0$

if $r’\leq p\leq\infty$ .

$S_{X,p}(\tau)$in$\mathrm{G}9$

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