94

数理解析研究所講究録
第2065巻 2018年 94-99

Rotation invariant norms on \mathb {R}^{2} and geometricconstants
Kichi‐Suke Saito, Naoto Komuro and Ryotaro Tanaka

1

introduction

In this paper, we mainly consider the James constants of rotation invariant norms on \mathbb{R}^{2}.
Let \Vert\cdot\Vert be a norm on \mathbb{R}^{2} , and let $\theta$\in(0,2 $\pi$) . Then \Vert\cdot| is said to be $\theta$ ‐rotation invariant
if the $\theta$ ‐rotation matrix

R($\thea$)=\left(\begin{ar y}{l \mathrm{c}\mathrm{o}\mathrm{s}$\thea$&-\mathrm{s}\mathrm{i}\mathrm{n}$\thea$\ mathrm{s}\mathrm{i}\mathrm{n}$\thea$&\infty\mathrm{s}$\thea$ \end{ar y}\right)

is an isometry on (\mathbb{R}^{2}, \Vert .
All norms on \mathb {R}^{2} are clearly $\pi$‐rotation invariant, and the$\theta$Euclidean norm is ‐rotation invariant for each $\theta$ \in (0,2 $\pi$) . For the study of James
constant, the class of $\pi$/ ‐rotation invariant norms are very important; and it is rich since
every symmetric absolute norms on \mathbb{R}^{2} , that is, a norm satisfying \Vert(|a|, |b|)\Vert=\Vert(a, b\Vert(b, a for each (a, b) , is $\pi$/2‐rotation invariant.
Now let X be a Banach space. Then S_{X} denotes the unit sphere of X . The James=constant J(X) of X was introduced in 1990 by Gao and Lau [3] as a measure of thesquareness of the unit ball. Namely, we define J(X) byJ(X)=\displaystyle \sup\{\mathrm{m}\mathrm{n}\{\Vert x+y \Vert x-y : x, y\in S_{X}\}.
It is known that J(X) has the following properties:(i) \sqrt{2}\leq J(X)\leq 2([3]) .
(ii) If H is a Hilbert space, then J(H)=\sqrt{2}.

(iii) If \dim X\geq 3 , then J(X)=\sqrt{2} implies that X is a Hilbert space ([5]); and hence
J(X)=\sqrt{2} if and only if X is a Hilbert space provided that \dim X\geq 3.
(iv) J(X)<2 if and only if X is uniformly non‐square, that is, there exists a $\delta$>0 suchthat \displaystyle \min\{\Vert x+y \Vert x-y<2(1- $\delta$) whenever x, y\in S_{X}.

We here emphasize that (iii) does not hold in the two‐dimensional case. Indeed, if weconsider the norm on \mathbb{R}^{2} whose unit sphere is a regular octagon, then its James constantis \sqrt{2} though it is clearly not a Hilbert space. Actually, this fact comes from the following
more general result by Gao and Lau [3, Proposition 2.8].

Proposition 1.1 (Gao and Lau [3]). Let \Vert\cdot\Vert be a $\pi$/4 ‐rotation invariant norm on \mathbb{R}^{2}.
Then J((\mathbb{R}^{2}, \Vert.
=\sqrt{2}.

95Thus, in fact, there are many non‐Hilbert two‐dimensional spaces with James constant\sqrt{2}.

The purpose of this paper is to consider the converse to the above proposition. We
present a partial converse by using the notion of rotation invariant norms on \mathb {R}^{2} . To dothis, the calculation formula for the James constants of $\pi$/2‐rotation rotation invariant
norms on \mathbb{R}^{2} is given. Moreover, we also give construction of $\pi$/ ‐rotation invariant norms
using certain convex functions on the unit interval.

2

James constants of $\pi$/2‐rotation invariant norms

We start this section with the following result of Komuro, Saito and Mitani [4] which
provides an important characterization of $\pi$/2‐rotation invariant norms on \mathbb{R}^{2} . Recall

that an element x of a normed space is said to be isosceles orthogonal to another elementy , denoted by x1_{I}y , if ||x+y\Vert=\Vert x-yTheorem 2.1. Let \Vert\cdot||\in N_{2} . Then the following are equivalent.

(i) \Vert\cdot\Vert is $\pi$/2 ‐rotation invariant.
(ii) x\perp_{I}y if and only if \langle x, y\rangle=0 whenever \Vert x\Vert=\Vert y\Vert=1.
In which cases, x\perp_{I}R( $\pi$/2)x for each x.

Now let $\Psi$_{2} be the set of all convex functions $\psi$ on [0 , 1 ] satisfying \displaystyle \max\{1-t, t\} \leq$\psi$(t)\leq 1 . For each $\psi$\in$\Psi$_{2} , the formula|(a,b)\Vert_{$\psi$}=\left\{ begin{ar ay}{l} (|a+|b)$\psi$(\frac{|b}{|a+|b})&( a,b)\neq(0, ) \ 0&( a,b)=(0, ) \end{ar ay}\right.
defines an absolute normalized norm on \mathbb{R}^{2} , that is, \Vert(a, b)\Vert_{ $\psi$}=\Vert(|a|, |b|)\Vert_{ $\psi$} for each (a, b)

and \Vert(1,0)\Vert_{ $\psi$}= \Vert(0,1)\Vert_{ $\psi$} . Let AN_{2} be the set of all absolute normalized norms on \mathbb{R}^{2}.

Then, the correspondence $\psi$\mapsto\Vert\cdot| _{ $\psi$} gives a bijection from $\Psi$_{2} onto AN_{2} ; see [2, 9].

Using convex functions in $\Psi$_{2} , we can construct more general two‐dimensional normed
spaces. Namely, for each pair $\varphi$, $\psi$\in$\Psi$_{2} , let | \cdot\Vert_{ $\varphi,\ \psi$} be the norm on \mathbb{R}^{2} given by\Vert(a,b)\Vert_{$\varphi,\ psi$}=\left\{ begin{ar ay}{l \Vert(a,b)\Vert_{$\varphi$}(ab\geq0)\ |(a,b)\Vert_{$\varphi$}(ab\geq0) \end{ar ay}\right.
The space (\mathbb{R}^{2}, \Vert\cdot\Vert_{ $\varphi,\ \psi$}) is called a Day‐James space, and is denoted by l_{ $\varphi,\ psi$}^{2} ; see [8].

In fact, the class of Day‐James spaces are essential among all two‐dimensional real

normed spaces in the following sense.

Proposition 2.2 (Alonso [1]). Every two‐dimensional real normed space is isometricallyisomorphic to some Day‐James space.

For each $\psi$\in$\Psi$_{2} , let \tilde{ $\psi$}(t)= $\psi$(1-t) for each t\in[0, 1 ] . Then \tilde{ $\psi$}\in$\Psi$_{2}.

The following proposition shows that the Day‐James space \el_{$\psi$,\tilde{$\psi$} ^{2} is always $\pi$/2‐rotation

invariant for any $\psi$\in$\Psi$_{2}.

96Proposition 2.3. Let $\psi$\in$\Psi$_{2} . Then \Vert\cdot\Vert_{ $\psi$,\tilde{ $\psi$} is $\pi$/2 ‐rotation invariant.

We have a refinement of Proposition 2.2 for the case of $\pi$/2‐rotation invariant norms

on\mathbb{R}^{2}.

Theorem 2.4 ([6]). Let | \cdot\Vert be a $\pi$/2 ‐rvtation invariant norm on \mathbb{R}^{2} . Then there esists
a $\psi$\in$\Psi$_{2} satisfying the following conditions.

(i) (\mathbb{R}^{2}, \Vert .

is isometrically isomorphic to P_{ $\psi$,\tilde{ $\psi$} .

(ii) | \cdot\Vert\in N_{2}( $\theta$) if and only if \Vert\cdot\Vert_{ $\psi$,\tilde{ $\psi$} \in N_{2}( $\theta$) for each $\theta$\in[0, 2 $\pi$].
Hence, the Day‐James space of the form P_{ $\psi$,\tilde{ $\psi$} are essential among ffi $\pi$/2‐rotation

invariant normed spaces.
We now consider the class of twodimensional normed spaces with James constant \sqrt{2}.
Let X be a normed space. For each x\in S_{X} , let$\beta$(x)=\displaystyle \sup\{\min\{\Vert x+y \Vert x-y : y\in S_{X}\}.

Then J(X)\displaystyle \sup\{ $\beta$(x) : x \in S_{X}\} . By [3, Lemma 2.2], if y \in S_{X} and x \perp_{I} y then\Vert x+y\Vert=\Vert x-y\Vert= $\beta$(x)The following is the $\pi$/2‐rotation invariant analogue of [7, Theorem 1].
=.

Proposition 2.5. Let $\psi$\in$\Psi$_{2} . Then

J(l_{ $\psi$,\tilde{ $\psi$} ^{2})=\displaystyle \max\{_{0}\max_{\leq $\iota$\leq 1/2}\frac{2-2t}{ $\psi$(t)} $\psi$(\frac{1}{2-2t}),\max\frac{2t}{ $\psi$(t)} $\psi$(\frac{2t-1}{2t})\}.
In what follows, for each $\psi$ , let

f_{ $\psi$}(t)=\displaystyle \frac{2-2t}{ $\psi$(t)} $\psi$(\frac{1}{2-2t})

andg_{ $\psi$}(t)=\displaystyle \frac{2t}{ $\psi$(t)} $\psi$(\frac{2t-1}{2t}),respectively. The following lemma is simple, but important.
Lemma 2.6. Let $\psi$\in$\Psi$_{2} . Thenf_{ $\psi$}(t)g_{ $\psi$}(\displaystyle \frac{1}{2-2t}) =2
for each t\in[0 , 1/2].

Combining the preceding lemma with Proposition 2.5, we have the following improve‐ment.

Theorem 2.7. Let $\psi$\in$\Psi$_{2} . Then

J(\displayst le\el_{$\psi$,\overline{$\psi$}^{2})=\max\{ max_{0\leqt\leq1/2}h(t),\frac{2}0\leqt\leq1/2\mathrm{ }\dot{\mathrm{ }f_{$\psi$}(t)\}.

Using this formula, we can characterize the $\pi$/2‐rotation invariant norms with James

constant \sqrt{2} as follows.

97

Theorem 2.8. Let $\psi$\in$\Psi$_{2} . Then the following are equivalent.

(i) J(\ell_{ $\psi$,\tilde{ $\psi$} ^{2})=\sqrt{2}.

(ü) h(t)=\sqrt{2} for each t\in[0, 1/2] and g_{ $\psi$}(t)=\sqrt{2} for each i\in[1/2, 1].
(iii) h(t)=\sqrt{2} for each l\in[0 , 1/2].
(iv) g_{ $\psi$}(t)=\sqrt{2} for each t\in[1/2, 1].

(v) \Vert\cdot\Vert_{ $\psi$,\tilde{ $\psi$} \in N_{2}( $\pi$/4) .
As a consequence of Theorems 2.4 and 2.8, we have the following partial converse toProposition 1.1, which is our aim in this section.
Theorem 2.9. Let \Vert\cdot\Vert\in N_{2}( $\pi$/2) . Then J((\mathbb{R}^{2}, \Vert3=\sqrt{2} if and only if \Vert\in N_{2}( $\pi$/4) .

Construction of $\pi$/ ‐rotation invariant norms=\sqrt{2} characterizes
As it was shown in the preceding section, the equality J((\mathbb{R}^{2}, ||.
the $\pi$/ ‐rotation invariant norms among the $\pi$/2‐rotation invariant norms. The aim of
this section is to study the structure of $\pi$/4-‐rotation invariant norms via some propertiesof certain convex functions on the unit interval.

Let DJ=. \Vert_{ $\varphi,\ \psi$} : $\varphi$, $\psi$\in$\Psi$_{2}\} , and let DJ( $\theta$) be the set of all elements in DJ which
are $\theta$‐rotation invariant. Then the set DJ is obviously in a one‐to‐one correspondence with$\Psi$_{2}\mathrm{x}$\Psi$_{2} . Moreover, if \Vert\cdot\Vert_{ $\varphi,\ \psi$}\in DJ( $\pi$/2) then $\psi$=\tilde{ $\varphi$}. This and Proposition 2.3 together
show that the set DJ( $\pi$/2) is in a one‐to‐one correspondence with the set \{( $\psi$,\tilde{ $\psi$}) : $\psi$\in$\Psi$_{2}\}which can be identified with $\Psi$_{2} . These observations are summarized as follows.

Proposition 3.1. The map $\Psi$_{2}\ni $\psi$\mapsto\Vert\cdot\Vert_{ $\psi$,\tilde{ $\psi$} \in DJ( $\pi$/2) is bijective.
Now let$\Gamma$=\displaystyle \{ $\psi$\in$\Psi$_{2}:\max\{1-(1-\frac{1}{\sqrt{2} )t, \frac{1}{\sqrt{2} +(1-\frac{1}{\sqrt{2} )t\}\leq $\psi$(t)\}.
Then we have the fonowing.
Proposition 3.2. Let $\psi$\in$\Psi$_{2} . If \Vert\cdot\Vert_{ $\psi$,\tilde{ $\psi$} \in DJ( $\pi$/4) , then$\psi$^{\mathrm{b} (t)=(1+(\displaystyle \sqrt{2}-1)t) $\psi$(\frac{t}{\sqrt{2}+(2-\sqrt{2})t})defines an element of $\Gamma$.

Remark 3.3. The preceding proposition is an extension of [5, Lemma 3.4] which statesthat \Vert\cdot\Vert_{ $\psi$}\in AN_{2}\cap N_{2}( $\pi$/4) implies $\psi$^{\mathrm{b} \in$\Gamma$^{\mathcal{S} = $\Gamma$\cap$\Psi$_{2}^{S}.

The following can be viewed as the “converse”’ of Proposition 3.2.

98

Proposition 3.4. Let $\psi$\in $\Gamma$ . Then$\psi$^{\mathrm{n}_(t)=}\left\{begin{ar y}{l (1-2\sqrt{2}) $\psi$(\frac{\sqrt{2} 1-(2\sqrt{2})l &(t\in[0,1/2]),\ (\sqrt{2}-1)(+\sqrt{2})$\psi$(\frac{2t-1}{(\sqrt{2}-1)(+\sqrt{2}) &(t\in[1/2,]) \end{ar y}\right.

defines an element of $\Psi$_{2} such that | \cdot\Vert_{ $\psi$\#,\overline{ $\psi$\#}}\in DJ( $\pi$/4) .

We now present the main theorem in this section which extends [5, Theorem 3.8] to$\pi$/4‐rotation invariant Day‐James norms.
Theorem 3.5. The following hold.

(i) ($\psi$^{\mathrm{b} )^{\#}= $\psi$ for each $\psi$\in$\Psi$_{2} with \Vert\cdot\Vert_{ $\psi$,\tilde{ $\psi$} \in DJ( $\pi$/4) .

(ii) ($\psi$^{\#})^{\mathrm{b} = $\psi$ for each $\psi$\in $\Gamma$.

(iii) The map $\Gamma$\ni $\psi$\mapsto\Vert\cdot\Vert_{ $\psi \iota$,\overline{ $\psi$\#}}\in DJ( $\pi$/4) is bijective.
The preceding theorem, together with Propositions 3.2 and 3.4, provide a specific way

to construct all $\pi$/4-‐rotation invariant norms on \mathbb{R}^{2}.

References[1] J. Alonso, Any two‐dimensional normed space is a generalized Day‐James space, J.
Inequal. Appl., 2011, 2011:2, 3 pp.

[2] $\Gamma$. $\Gamma$ . Bonsall and J. Duncan, Numerical ranges II, Cambridge University Press,Cambridge, 1973.

[3] J. Gao and K.‐S. Lau, On the geometry of spheres in normed linear spaces, J. Aưt.
Math. Soc. Ser. A, 48 (1990), 101‐112.
[4] N. Komuro, K.‐S. Saito and K.‐I. Mitani, Convex property of James and von
Neumann‐Jordan constant of absolute norms on \mathbb{R}^{2} , Proceedings of the 8th Interna‐tional Conference on Nonlinear Analysis and Convex Analysis, 301‐308, Yokohama
Publ., Yokohama, 2015.

[5] N. Komuro, K.‐S. Saito and R. Tanaka, On the class of Banach spaces with James
constant \sqrt{2} , 289 (2016), 1005‐1020.
[6] N. Komuro, K.‐S. Saito and R. Tanaka, On the class of Banach spaces with James
constant \sqrt{2} : Part II, Mediterr. J. Math., 13 (2016), 4039‐4061.
[7] K.‐I. Mitani and K.‐S. Saito, The James constant of absolute norms on \mathbb{R}^{2} , J. Non‐linear Convex Anal., 4 (2003), 39k4l0.
[8] W. Nilsrakoo and S. Saejung, The James constant of normalized norms on \mathbb{R}^{2} , J.
Inequal. Appl. 2006, Art. ID 26265, 12 pp.

99[9] K.‐S. Saito, \mathrm{M} , Kato and Y. Takahashi, Von Neumann‐Jordan constant of absolutenormalized nonns on \mathb {C}^{2} , J. Math. Anal. Appl., 244 (2000), 515‐532.
Kichi‐Suke Saito

Department of Mathematical Sciences,
Institute of Science and Technology,
Niigata University,
Niigata 950‐2181, Japan
Bmail: saito@math.sc.niigata‐u.ac.jp
Naoto Komuro

Department of Mathematics,
Hokkaido University of Education, Asahikawa Campus,
Asahikawa 070\mapsto 8621 , JapanE‐‐mail: komuro@asa.hokkyodai.ac.jp
Ryotaro Tanaka
Faculty of Mathematics,Kyushu University,Fukuoka 819‐0395, JapanE‐‐mail: r‐tanaka@math.kyushu‐u.ac.jp