Then we calculateγX,ψ(t)for some Banach spaces

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OF THE UNIT BALL OF BANACH SPACES

HIROYASU MIZUGUCHI

Communicated by Vasile Brnzanescu

In 2009, Mitani and Saito introduced and studied a geometric constant γX,ψof a Banach spaceX, by using the notion ofψ-ditect sum. Fort∈[0,1], the constant γX,ψ(t)is dened as a supremum taken over all elements in the unit sphere of X.

In this paper, we obtain that, for a Banach space which has a predual Banach space, the supremum can be taken over all extreme points of the unit ball. Then we calculateγX,ψ(t)for some Banach spaces.

AMS 2010 Subject Classication: 46B20.

Key words: absolute normalized norm,ψ-direct sum, extreme point, von Neumann- Jordan constant, James type constant, von Neumann-Jordan type constant.

1. INTRODUCTION

There are several constants dened on Banach spaces such as the James constant [4] and von Neumann-Jordan constant [3]. It has been shown that these constants are very useful in the study of geometric structure of Banach spaces.

Throughout this paper, letX be a Banach space withdimX≥2. ByS_X and B_X, we denote the unit sphere and the unit ball of X, respectively. The von Neumann-Jordan constant is dened by

C_{N J}(X) = sup

kx+yk²+kx−yk²

2(kxk²+kyk²) :x, y∈X,(x, y)6= (0,0)

(Clarkson [3]), where the supremum can be taken over allx∈SX andy ∈BX. This constant has been considered in many papers ([3, 5, 8, 15, 17, 18] and so on). It is known that

(i) For any Banach spaceX,1≤C_{N J}(X)≤2.

(ii) X is a Hilbert space if and only if C_{N J}(X) = 1 ([5]).

(iii) X is uniformly non-square if and only if CN J(X)<2 ([17]).

REV. ROUMAINE MATH. PURES APPL. 60 (2015), 1, 5970

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We note that the von Neumann-Jordan constant C_{N J}(X)is reformulated as

CN J(X) = sup

kx+tyk²+kx−tyk²

2(1 +t²) :x, y∈SX,0≤t≤1

.

In 2006, the functionγX from[0,1]into[0,4]was introduced by Yang and Wang [21]:

γX(t) = sup

kx+tyk²+kx−tyk²

2 :x, y∈SX

.

This function is useful to calculate the von Neumann-Jordan constant C_{N J}(X) for some Banach spaces. In fact, they computedC_{N J}(X)for X being Day-James spaces `∞-`1 and `2-`1 by using the function γX. In [11], Mitani and Saito introduced a geometrical constantγ_X,ψ of a Banach space, by using the notion ofψ-direct sum.

Recall that a norm k · kon C² is said to be absolute if k(z, w)k=k(|z|,|w|)k

for all (z, w) ∈ C², and normalized if k(1,0)k = k(0,1)k = 1. The family of all absolute normalized norms on C² is denoted by AN2. As in Bonsall and Duncan [2], AN₂ is in a one-to-one correspondence with the family Ψ₂ of all convex functions ψ on [0,1] with max{1−t, t} ≤ ψ(t) ≤ 1 for all 0 ≤ t ≤1. Indeed, for anyk·k ∈AN2 we putψ(t) =k(1−t, t)k. Thenψ∈Ψ2. Conversely, for all ψ∈Ψ₂ let

k(z, w)k_ψ =







(|z|+|w|)ψ

|w|

|z|+|w|

if (z, w)6= (0,0),

0 if (z, w)6= (0,0).

Then k · k_ψ ∈ AN₂, and ψ(t) = k(1−t, t)k_ψ (cf. [15]). The functions corresponding to the`p-normsk·k_ponC²are given byψp(t) ={(1−t)^p+t^p}^1/p if 1≤p <∞, and ψ∞(t) = max{1−t, t}if p=∞.

Takahashi, Kato and Saito [19] used the previous fact to introduce the notion of ψ-direct sum of Banach spaces X and Y as their direct sum X⊕Y equipped with the norm

k(x, y)k_ψ =k(kxk,kyk)k_ψ ((x, y)∈X⊕Y).

We denote byX⊕_ψY the direct sumX⊕Y with this norm. This notion has been studied by several authors (cf. [6, 7, 10, 14]).

For a Banach space X and ψ∈Ψ2, the function γX,ψ on[0,1] is dened by

γ_X,ψ(t) = sup{k(x+ty, x−ty)k_ψ :x, y∈S_X}. Mitani and Saito [11] showed that

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Proposition 1.1 ([11]).

(1) For any Banach space X, ψ∈Ψ₂ and t∈[0,1], 2ψ

1−t 2

≤γ_X,ψ(t)≤2(1 +t)ψ 1

2

. (2) For a Banach space X, ψ∈Ψ2 and t∈[0,1],

γX,ψ(t) = sup{k(x+ty, x−ty)k_ψ :x, y∈BX}.

(3) Let ψ ∈ Ψ₂ with ψ 6= ψ∞. Then a Banach space X is uniformly non-square if and only if γX,ψ(t) < 2(1 +t)ψ(1/2) for any (or some) t with 0< t≤1.

They also gave the value of γ_X,ψ when X is a Hilbert space and an `_p- space, and obtained a sucient condition for uniform normal structure of Ba- nach spaces in terms ofγ_X,ψ.

Our aim in this paper is to study some properties ofγ_X,ψ. In particular, we prove that for a Banach spaceXwith a predual Banach spaceX∗, the function γ_X,ψ(t)can be calculated as the supremum taken over all extreme points of the unit ball. Then we calculateγ_X,ψ(t)for X being Day-James spaces`∞-`₁ and

`₂-`₁.

2. SOME PROPERTIES OF γX,ψ

We easily obtain the following properties of γ_X,ψ.

Proposition 2.1. Let X be a Banach space, and letψ∈Ψ₂. Then (1) γ_X,ψ(t) is a non-decreasing function;

(2) γX,ψ(t) is a convex function;

(3) γ_X,ψ(t) is continuous on[0,1]. (4) The function

γX,ψ(t)−γX,ψ(0) is non-decreasing on (0,1]. t

Proof. (1) Let 0≤t₁ ≤t₂ ≤1. Take anyx, y ∈S_X. Since ^t_t¹₂y∈B_X, by Proposition 1.1 (2), we have

k(x+t₁y, x−t₁y)k_ψ =

x+t₂·t₁

t₂y, x−t₂·t₁ t₂y

ψ

≤γ_X,ψ(t₂).

Thus, we obtain γ_X,ψ(t₁)≤γ_X,ψ(t₂).

(2) Let t₁, t₂ ∈[0,1] and λ∈(0,1). Then, from the convexity of k · k_ψ ∈ AN2, we obtain

γ_X,ψ((1−λ)t₁+λt₂))≤(1−λ)γ_X,ψ(t₁) +λγ_X,ψ(t₂).

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(3) Since (2) implies that γ_X,ψ(t) is continuous on (0,1), it suces to show that γ_X,ψ(t) is continuous att= 0 andt= 1. From Proposition 1.1 (1), it follows thatγX,ψ(t)−γX,ψ(0)≤2tψ(1/2)for anyt∈(0,1). Thus,γX,ψ(t)is continuous att= 0.

Take any t∈(0,1)and anyx, y∈SX. Sincetx∈BX, by Proposition 1.1 (2), one can have that

tk(x+y, x−y)k_ψ =k(tx+ty, tx−ty)k_ψ ≤γ_X,ψ(t)

and hence, tγ_X,ψ(1) ≤ γ_X,ψ(t). Thus, we obtain γ_X,ψ(1)−γ_X,ψ(t) ≤ (1− t)γ_X,ψ(1), which completes the proof.

(4) This is an easy consequence of (2).

In [21], the sucient condition of uniform smoothness was obtained in terms of γX. Related to this result, we obtain the following

Proposition 2.2. Let ψ ∈ Ψ₂. Assume that ψ takes the minimum at t= 1/2. Then, a Banach space X is uniformly smooth if

t→0lim₊

γ_X,ψ(t)−γ_X,ψ(0)

t = 0.

Proof. Put M = 1/min0≤t≤1ψ(t). Then, for any t ∈ (0,1] and any x, y∈S_X, we have

kx+tyk+kx−tyk

2 −1 = k(x+ty, x−ty)k₁

2 −1

≤ M γX,ψ(t)

2 −1

= M 2

γ_X,ψ(t)−k(1,1)k₁ M

= M

2 (γ_X,ψ(t)− k(1,1)k_ψ)≤γ_X,ψ(t)−γ_X,ψ(0), which implies ρX(t)≤γX,ψ(t)−γX,ψ(0). Thus, we obtain

t→0lim₊ ρ_X(t)

t ≤ lim

t→0₊

γ_X,ψ(t)−γ_X,ψ(0)

t = 0

and then X is uniformly smooth.

An element x ∈ S_X is called an extreme point of B_X if y, z ∈ S_X and x= (y+z)/2impliesx=y=z. The set of all extreme points ofBX is denoted by ext(B_X). There exists some innite-dimensional Banach spaces whose unit ball has no extreme point. However, from the Banach-Alaoglu Theorem and

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Krein-Milman Theorem, we have that for any Banach space, the unit ball of the dual space is the weakly^∗ closed convex hull of its set of extreme points.

For ψ∈Ψ2, the dual functionψ^∗ of ψis dened by ψ^∗(s) = sup

t∈[0,1]

(1−s)(1−t) +st ψ(t)

for s∈[0,1]. Then we have ψ^∗∈Ψ₂ and thatk · k_ψ^∗ is the dual norm ofk · k_ψ. It is easy to see that ψ^∗∗=ψ. Let Y, Z be Banach spaces. Then according to [10], the dual ofY ⊕_ψZ is isomorphic toY^∗⊕_ψ^∗Z^∗.

Suppose that X is a Banach space which has the predual Banach space X∗. Then the unit ballBX is the weakly^∗ closed convex hull of ext(BX), and the direct sum X⊕_ψX is isomorphic to the dual of X∗⊕_ψ^∗X∗.

Theorem 2.3. Let X be a Banach space with the predual Banach space.

Then

γX,ψ(t) = sup{k(x+ty, x−ty)k_ψ :x, y∈ext(BX)}

for any ψ∈Ψ₂ and any t∈[0,1].

Proof. Letψ∈Ψ2andt∈[0,1]. Take arbitrary elementsx, y∈BX. It follows fromy∈B_X = co^w∗(ext(B_X))that there exists a net{y_α}inco(ext(B_X)) which weakly^∗ converges to y. Since the net {(x+ty_α, x−ty_α)}weakly^∗ converges to (x+ty, x−ty)∈X⊕_ψX, we obtain

k(x+ty, x−ty)k_ψ ≤lim

α

k(x+ty_α, x−ty_α)k_ψ

≤sup

α

k(x+ty_α, x−ty_α)k_ψ

= sup{k(x+tv, x−tv)k_ψ :v∈co(ext(B_X))}. For any v ∈ co(ext(BX)), since x ∈ BX = co^w∗(ext(BX)), as in the preceding paragraph, we have

k(x+tv, x−tv)k_ψ ≤sup{k(u+tv, u−tv)k_ψ :u∈co(ext(BX))}. Hence, we obtain

k(x+ty, x−ty)k_ψ ≤sup{k(u+tv, u−tv)k_ψ :u, v∈co(ext(BX))}. On the other hand, from the convexity of k · k_ψ ∈AN₂, we directly have

sup{k(x+ty, x−ty)k_ψ :x, y∈co(ext(BX))}

= sup{k(x+ty, x−ty)k_ψ :x, y∈ext(BX)}. Thus, we obtain this theorem.

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In [16], Takahashi introduced the James and von Neumann-Jordan type constants of Banach spaces. For t ∈ [−∞,∞) and τ ≥ 0, the James type constant is dened as

JX,t(τ) =









 sup

(kx+τ yk^t+kx−τ yk^t 2

1/t

:x, y∈S_X )

if t6=−∞,

sup{min(kx+τ yk,kx−τ yk) :x, y∈S_X} if t=−∞

(cf. [20, 22]). The von Neumann-Jordan type constant is dened as Ct(X) = sup

JX,t(τ)²

1 +τ² : 0≤τ ≤1

.

Forq∈[1,∞)andt∈[0,1], it is easy to see thatJ_X,q(t) = 2^−1/qγ_X,ψ_q(t). Thus, we have the following results on the James and von Neumann-Jordan type constants.

Corollary 2.4. Let Xbe a Banach space with the predual Banach space.

(1) For any q ∈[1,∞) and any t∈[0,1], JX,q(t) = sup

(

kx+tyk^q+kx−tyk^q 2

1/q

:x, y∈ext(BX) )

(2) For any q ∈[1,∞), C_q(X) = sup

((kx+tyk^q+kx−tyk^q)^2/q

2^2/q(1 +t²) :x, y∈ext(B_X),0≤t≤1 )

.

In particular, one can easily has

ρ_X(t) =J_X,1(t)−1 = γX,ψ1(t)

2 −1

for any t∈[0,1], and

CN J(X) =C2(X) = sup

γ_X,ψ₂(t)²

2(1 +t²) : 0≤t≤1

. Hence, we obtain

Corollary 2.5. Let Xbe a Banach space with the predual Banach space.

Then,

ρ_X(t) = sup

kx+tyk+kx−tyk

2 −1 :x, y∈ext(B_X)

for all t∈[0,1], and CN J(X) = sup

kx+tyk²+kx−tyk²

2(1 +t²) :x, y∈ext(BX), 0≤t≤1

.

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3. EXAMPLES

In this section, we calculate γX,ψ(t) for some two-dimensional Banach spaces. Then we mention a geometric constant which can not be expressed by γ_X,ψ(t).

For p, q with 1 ≤p, q ≤ ∞, the Day-James space `p-`q is dened as the space R² with the norm

k(x₁, x₂)k_p,q=

( k(x₁, x₂)k_p if x₁x₂ ≥0, k(x₁, x2)k_q if x1x2 ≤0.

Yang and Wang [21] calculated the von Neumann-Jordan constant of the Day- James spaces`∞-`1and`2-`1by using the notion ofγ_X(t). We computeγ_X,ψ(t) of these spaces for all ψ ∈ Ψ₂ and all t∈[0,1]. Remark that `∞-`₁ and `₂-`₁ have the predual spaces `1-`∞ and `2-`∞, respectively (cf. [13]). Thus, from Theorem 2.3, we obtain

γ_X,ψ(t) = sup{k(x+ty, x−ty)k_ψ :x, y∈ext(B_X)}

for X being`∞−`1 or`2−`1. We note that, for x, y∈ext(BX),k(x−ty, x+ ty)k_ψ does not necessarily coincide withk(x+ty, x−ty)k_ψ.

Example 3.1. LetXbe the Day-James space`∞−`₁,ψ∈Ψ2andt∈[0,1].

Then

γ_X,ψ(t) = (2 +t) max

ψ 1

2 +t

, ψ

1 +t 2 +t

. In particular, for q ∈[1,∞),

JX,q(t) =

1 + (1 +t)^q 2

1/q

and Cq(X) = {1 + (1 +t0)^q}^2/q 2^2/q(1 +t²₀) , wheret0 ∈(0,1)such that(1 +t0)^q−1(1−t0)−t0= 0.

Proof. It is easy to check

ext(BX) ={±(1,1), (±1,0), (0,±1)}.

By the denition of k · k_∞,1, we may consider k(x+ty, x−ty)k_ψ and k(x−ty, x+ty)k_ψ only in the following three cases.

Case 1. x= (1,0),y= (0,1). We have

kx+tyk_∞,1 =k(1, t)k_∞,1= 1 and kx−tyk_∞,1=k(1,−t)k_∞,1 = 1 +t.

Case 2. x= (1,1),y= (1,0). We have

kx+tyk_∞,1 =k(1 +t,1)k_∞,1 = 1 +t and

kx−tyk∞,1 =k(1−t,1)k∞,1 = 1.

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Case 3. x= (1,0),y= (1,1). We have

kx+tyk∞,1=k(1 +t, t)k∞,1 = 1 +t and

kx−tyk_∞,1 =k(1−t,−t)k_∞,1 = 1.

Thus, we obtain

γX,ψ(t) = max{k(1 +t,1)k_ψ,k(1,1 +t)k_ψ}

= (2 +t) max

ψ 1

2 +t

, ψ

1 +t 2 +t

,

and hence, forq ∈[1,∞),

JX,q(t) =

1 + (1 +t)^q 2

1/q

and

C_q(X) = sup

({1 + (1 +t)^q}^2/q

2^2/q(1 +t²) : 0≤t≤1 )

.

Let t₀ ∈(0,1)with(1 +t₀)^q−1(1−t₀)−t₀ = 0. Then the function {1 + (1 +t)^q}^2/q

2^2/q(1 +t²)

takes the maximum at t=t₀. This completes the proof.

To calculateγ_X,ψ(t)forX being the Day-James space`₂-`₁, we note that for any ψ ∈Ψ₂, if |z| ≤ |u| and |w| ≤ |v|, then k(z, w)k_ψ ≤ k(u, v)k_ψ, and if

|z|<|u|and|w|<|v|, thenk(z, w)k_ψ <k(u, v)k_ψ (cf. [2]). More results on the monotonicity of absolute normalized norms can be found in [12, 10, 19] and so on.

Example 3.2. Let X be the Day-James space `2-`1,ψ∈Ψ2 and t∈[0,1].

Then γ_X,ψ(t)

= (1 +t+p

1 +t²) max (

ψ

1 +t 1 +t+√

1 +t²

, ψ

√ 1 +t² 1 +t+√

1 +t²

!) .

In particular, for q ∈[1,∞),

J_X,q(t) = (1 +t)^q+ (1 +t²)^q/2 2

!1/q

and Cq(X) = 1 + 2^q/2 2

!2/q

.

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Proof. One can easily have that

ext(B_X) ={(x₁, x2) :x²₁+x²₂= 1, x1x2≥0}.

Let t∈ [0,1]. For θ₁, θ₂ with 0≤θ₁ ≤θ₂ ≤π/2, put x = (cosθ₁,sinθ₁) and y = (cosθ₂,sinθ₂). Then x+ty = (cosθ₁+tcosθ₂,sinθ₁+tsinθ₂) and x−ty = (cosθ1−tcosθ2,sinθ1−tsinθ2). Thus, we have

kx+tyk_2,1 =kx+tyk₂=p

1 +t²+ 2tcos(θ2−θ1) and hence, √

1 +t² ≤ kx+tyk_2,1 ≤1 +t. If sinθ₁ ≥tsinθ₂, then one has

kx−tyk_2,1 =kx−tyk₂=p

1 +t²−2tcos(θ2−θ1)≤p 1 +t². Hence, from the monotonicity of k · k_ψ ∈AN2,

k(x+ty, x−ty)k_ψ ≤ k(1 +t,p

1 +t²)k_ψ and

k(x−ty, x+ty)k_ψ ≤ k(p

1 +t²,1 +t)k_ψ. Suppose thatsinθ₁ < tsinθ₂. Then we have

kx−tyk_2,1=kx−tyk₁ = cosθ₁−tcosθ₂−sinθ₁+tsinθ₂. One can show that kx+tyk_2,1 +kx−tyk_2,1 ≤1 +t+√

1 +t². Indeed, putting e1= (1,0)and e2= (0,1), by the triangle inequality, we have

kx+tyk_2,1−p

1 +t²=kx+tyk_2,1− ke₁+te₂k_2,1

≤ k(x+ty)−(e₁+te₂)k_2,1

≤ kx−e₁k_2,1+tky−e₂k_2,1

= 1−cosθ₁+ sinθ₁+t(cosθ₂+ 1−sinθ₂)

= 1 +t− kx−tyk_2,1. On the other hand, we have already obtained that√

1 +t²≤ kx+tyk_2,1≤ 1 +t. Thus, by the monotonicity and convexity ofk · k_ψ ∈AN₂, we have

k(x+ty, x−ty)k_ψ ≤max{k(p

1 +t²,1 +t)k_ψ,k(1 +t,p

1 +t²)k_ψ} and

k(x−ty, x+ty)k_ψ ≤max{k(p

1 +t²,1 +t)k_ψ,k(1 +t,p

1 +t²)k_ψ}.

Therefore we obtain γ_X,ψ(t)≤max{k(p

1 +t²,1 +t)k_ψ,k(1 +t,p

1 +t²)k_ψ}.

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Finally, for e₁ = (1,0)and e₂ = (0,1), one has k(e₁+te₂, e₁−te₂)k_ψ =k(p

1 +t²,1 +t)k_ψ and

k(e₁−te₂, e₁+te₂)k_ψ =k(1 +t,p

1 +t²)k_ψ. Thus, we obtain

γ_X,ψ(t)

= max{k(p

1 +t²,1 +t)k_ψ,k(1 +t,p

1 +t²)k_ψ}

= (1 +t+p

1 +t²) max (

ψ

1 +t 1 +t+√

1 +t²

, ψ

√1 +t² 1 +t+√

1 +t²

!)

and hence, forq ∈[1,∞),

J_X,q(t) = (1 +t)^q+ (1 +t²)^q/2 2

!1/q

and

Cq(X) = sup

({(1 +t)^q+ (1 +t²)^q/2}^2/q

2^2/q(1 +t²) : 0≤t≤1 )

. Since the function

{(1 +t)^q+ (1 +t²)^q/2}^2/q 2^2/q(1 +t²) is increasing on the interval [0,1], one has

C_q(X) = (2^q+ 2^q/2)^2/q

2·2^2/q = (1 + 2^q/2)^2/q 2^2/q , as desired.

Although Theorem 2.3 holds, some geometric constants does not necessarily coincide with the supremum taken over all extreme points of the unit ball. We show a such example.

The constant CZ(X) = sup

kx+ykkx−yk

kxk²+kyk² :x, y∈X, (x, y)6= (0,0)

.

was introduced by Zbaganu [23]. As in the von Neumann-Jordan constant, this constant is reformulated as

CZ(X) = sup

kx+tykkx−tyk

1 +t² :x, y∈SX, 0≤t≤1

.

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Example 3.3. Let X be the Day-James space `∞-`₁. Then sup

kx+tykkx−tyk

1 +t² :x, y∈ext(B_X), 0≤t≤1

< C_Z(X).

Proof. According to [1], we have C_Z(X) = 5/4. On the other hand, as in Example 3.1, we obtain sup

kx+tykkx−tyk

1 +t² :x, y∈ext(B_X), 0≤t≤1

= max

0≤t≤1

1 +t 1 +t²

= 1

2(√ 2−1), and hence, this supremum is less than the Zbaganu constant CZ(X).

Remark 3.4. From [16], Zbaganu constant C_Z(X) coincide with the von Neumann-Jordan type constant C0(X).

We do not know whether, for anyqless than1, there exist a Banach space X in which the von Neumann-Jordan type constant C_q(X) does not coincide with the supremum taken over all extreme points of the unit ball BX.

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Received 29 October 2013 Niigata University,

Department of Mathematical Sciences, Graduate School of Science

and Technology, Niigata 950-2181, Japan mizuguchi@m.sc.niigata-u.ac.jp