A new geometrical constant of Banach spaces and the uniform normal structure

Ken-Ichi Mitani, Kichi-Suke Saito ^∗

A new geometrical constant of Banach spaces and the uniform normal structure

Abstract. We introduce and study a new geometrical constant 𝛾

of a Banach space 𝑋, by using the notion of 𝜓-direct sum given in [Y. Takahashi, M. Kato and K.

-S. Saito, J. Inequal. Appl. 7 (2002), 179-186]. At first, we characterize uniform non- squareness in terms of 𝛾

. Moreover, we consider Banach spaces having uniform normal structure.

2000 Mathematics Subject Classification: 46B20.

Key words and phrases: absolute norm, 𝜓-direct sum, uniformly non-square, uniform normal structure.

1. Introduction and Preliminaries. A norm ∥⋅∥ on ℂ ² is said to be absolute if

∥(∣𝑥∣, ∣𝑦∣)∥ = ∥(𝑥, 𝑦)∥

for all (𝑥, 𝑦) ∈ ℂ ² , and normalized if ∥(1, 0)∥ = ∥(0, 1)∥ = 1. The family of all absolute normalized norms on ℂ ² is denoted by 𝐴𝑁 2 . As in Bonsall and Duncan [3], 𝐴𝑁 2 is in a 1-1 correspondence with the family Ψ 2 of all continuous convex functions 𝜓 on [0, 1] with 𝜓(0) = 𝜓(1) = 1 and max {1 − 𝑡, 𝑡} ≤ 𝜓(𝑡) ≤ 1 for all 0 ≤ 𝑡 ≤ 1. Indeed, for any ∥ ⋅ ∥ ∈ 𝐴𝑁 ² we put 𝜓(𝑡) = ∥(1 − 𝑡, 𝑡)∥. Then 𝜓 ∈ Ψ ² . Conversely, for all 𝜓 ∈ Ψ ² let

∥(𝑥, 𝑦)∥ ^𝜓 =

⎧ 

⎨

 ⎩

( ∣𝑥∣ + ∣𝑦∣)𝜓 ( ∣𝑦∣

∣𝑥∣ + ∣𝑦∣

)

if (𝑥, 𝑦) ∕= (0, 0),

0 if (𝑥, 𝑦) = (0, 0).

Then ∥ ⋅ ∥ ^𝜓 ∈ 𝐴𝑁 ² , and 𝜓(𝑡) = ∥(1 − 𝑡, 𝑡)∥ ^𝜓 . The functions which correspond to the ℓ 𝑝 -norms ∥ ⋅ ∥ ^𝑝 on ℂ ² are 𝜓 𝑝 (𝑡) = {(1 − 𝑡) ^𝑝 + 𝑡 ^𝑝 } ^1/𝑝 if 1 ≤ 𝑝 < ∞, and 𝜓 _∞ (𝑡) = max {1 − 𝑡, 𝑡} if 𝑝 = ∞.

∗

The second author was supported in part by Grants-in-Aid for Scientific Research (No.

20540158), Japan Society for the Promotion of Science.

Recently, Takahashi, Kato and Saito [17] 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

∥(𝑥, 𝑦)∥ ^𝜓 = ∥(∥𝑥∥, ∥𝑦∥)∥ ^𝜓 (

(𝑥, 𝑦) ∈ 𝑋 ⊕ 𝑌 ) .

This notion has been studied by several authors (cf. [6, 9, 10, 14, 15, 17]).

In this paper, we introduce and study a new geometrical constant 𝛾 𝑋,𝜓 of a Banach space 𝑋 by using the notion of the 𝜓-direct sum. We first characterize the uniform non-squareness of Banach spaces in terms of 𝛾 𝑋,𝜓 . To do it we discuss the monotonicity of absolute norms on ℂ ² (Section 2). We also give the value of 𝛾 𝑋,𝜓

when 𝑋 is a Hilbert space and an ℓ 𝑝 space. Moreover, we consider Banach spaces having uniform normal structure.

Recall several geometrical constants of Banach spaces (cf. [1, 2, 4, 7]). The modulus of convexity of a Banach space 𝑋 is the function 𝛿 𝑋 : [0, 2) → [0, 1] defined by

𝛿 𝑋 (𝜀) = inf {

1 − ∥𝑥 + 𝑦∥

2 : 𝑥, 𝑦 ∈ 𝑆 ^𝑋 , ∥𝑥 − 𝑦∥ ≥ 𝜀 }

,

where 𝑆 𝑋 is the unit sphere of 𝑋. If 𝛿 𝑋 (𝜀) > 0 for some 𝜀 > 0, then 𝑋 is called uniformly non-square. The modulus of smoothness of a Banach space 𝑋 is the function 𝜌 𝑋 : [0, 1] → [0, 1] defined by

𝜌 𝑋 (𝑡) = sup

{ ∥𝑥 + 𝑡𝑦∥ + ∥𝑥 − 𝑡𝑦∥

2 − 1 : 𝑥, 𝑦 ∈ 𝑆 ^𝑋 }

(cf. [13]). The function 𝛾 𝑋 from [0, 1] → [0, 4] was introduced by Yang and Wang [18]:

𝛾 𝑋 (𝑡) = sup

{ ∥𝑥 + 𝑡𝑦∥ ² + ∥𝑥 − 𝑡𝑦∥ ²

2 : 𝑥, 𝑦 ∈ 𝑆 ^𝑋

}

(0 ≤ 𝑡 ≤ 1).

2. Monotonicity of absolute norms on ℂ ² . In this section we discuss the monotonicity of absolute normalized norms on ℂ ² . It is known that for any 𝜓 ∈ Ψ ² , if ∣𝑧∣ ≤ ∣𝑢∣ and ∣𝑤∣ ≤ ∣𝑣∣, then ∥(𝑧, 𝑤)∥ ^𝜓 ≤ ∥(𝑢, 𝑣)∥ ^𝜓 , and if ∣𝑧∣ < ∣𝑢∣ and ∣𝑤∣ < ∣𝑣∣, then ∥(𝑧, 𝑤)∥ ^𝜓 < ∥(𝑢, 𝑣)∥ ^𝜓 (cf. [3]). In [17], Takahashi, Kato and Saito showed the following.

Proposition 2.1 ([17], Proposition 2a) Let 𝜓 ∈ Ψ ² . Then the following are equivalent:

(i) 𝜓(𝑡) > 𝑡 for all 𝑡 with 0 < 𝑡 < 1.

(ii) 𝜓(𝑡)/𝑡 is strictly decreasing on (0, 1].

(iii) If ∣𝑧∣ < ∣𝑢∣ and ∣𝑤∣ ≤ ∣𝑣∣, then ∥(𝑧, 𝑤)∥ ^𝜓 < ∥(𝑢, 𝑣)∥ ^𝜓 .

Proposition 2.2 ([17], Proposition 2b) Let 𝜓 ∈ Ψ ² . Then the following are equivalent:

(i) 𝜓(𝑡) > 1 − 𝑡 for all 𝑡 with 0 < 𝑡 < 1.

(ii) 𝜓(𝑡)/(1 − 𝑡) is strictly increasing on [0, 1).

(iii) If ∣𝑧∣ ≤ ∣𝑢∣ and ∣𝑤∣ < ∣𝑣∣, then ∥(𝑧, 𝑤)∥ ^𝜓 < ∥(𝑢, 𝑣)∥ ^𝜓 .

We shall give refinements of the previous results.

Theorem 2.3 Let 𝜓 ∈ Ψ ² and let 𝑡 0 with 1/2 ≤ 𝑡 ⁰ ≤ 1. Then the following are equivalent:

(i) 𝜓(𝑡) > 𝑡 for all 𝑡 with 0 < 𝑡 < 𝑡 0 , and 𝜓(𝑡) = 𝑡 for all 𝑡 with 𝑡 0 ≤ 𝑡 ≤ 1.

(ii) 𝜓(𝑡)/𝑡 is strictly decreasing on (0, 𝑡 0 ), and 𝜓(𝑡)/𝑡 = 1 for all 𝑡 with 𝑡 0 ≤ 𝑡 ≤ 1.

(iii) Let ∣𝑧∣ < ∣𝑢∣. If _{∣𝑢∣+∣𝑤∣} ^∣𝑤∣ < 𝑡 0 , then ∥(𝑧, 𝑤)∥ ^𝜓 < ∥(𝑢, 𝑤)∥ ^𝜓 , and if _{∣𝑢∣+∣𝑤∣} ^∣𝑤∣ ≥ 𝑡 ⁰ , then ∥(𝑧, 𝑤)∥ ^𝜓 = ∥(𝑢, 𝑤)∥ ^𝜓 .

Proof (i) ⇒(ii). Assume (i). Let 0 < 𝑠 < 𝑡 < 𝑡 ⁰ . By the convexity of 𝜓, 𝜓(𝑡) ≤ 1 − 𝑡

1 − 𝑠 𝜓(𝑠) + 𝑡 − 𝑠 1 − 𝑠 𝜓(1).

Hence we have 𝜓(𝑠)

𝑠 − 𝜓(𝑡)

𝑡 ≥ 𝜓(𝑠)

𝑠 − 1 𝑡

{ 1 − 𝑡

1 − 𝑠 𝜓(𝑠) + 𝑡 − 𝑠 1 − 𝑠 ⋅ 1

}

= 𝑡 − 𝑠

𝑠𝑡(1 − 𝑠) {𝜓(𝑠) − 𝑠} > 0.

Thus 𝜓(𝑡)/𝑡 is strictly decreasing on (0, 𝑡 0 ).

(ii) ⇒(iii). Assume (ii). Let ∣𝑧∣ < ∣𝑢∣ and suppose that _{∣𝑢∣+∣𝑤∣} ^∣𝑤∣ < 𝑡 0 . In the case of 𝑡 0 > _{∣𝑧∣+∣𝑤∣} ^∣𝑤∣ > _{∣𝑢∣+∣𝑤∣} ^∣𝑤∣ we have

∥(𝑧, 𝑤)∥ ^𝜓 = ∣𝑤∣ ⋅ 𝜓( _{∣𝑧∣+∣𝑤∣} ^∣𝑤∣ )

∣𝑤∣

∣𝑧∣+∣𝑤∣

< ∣𝑤∣ ⋅ 𝜓( _{∣𝑢∣+∣𝑤∣} ^∣𝑤∣ )

∣𝑤∣

∣𝑢∣+∣𝑤∣

= ∥(𝑢, 𝑤)∥ ^𝜓 .

In the case of _{∣𝑧∣+∣𝑤∣} ^∣𝑤∣ ≥ 𝑡 ⁰ > _{∣𝑢∣+∣𝑤∣} ^∣𝑤∣ we have

∥(𝑧, 𝑤)∥ ^𝜓 = ∣𝑤∣ ⋅ 𝜓( _{∣𝑧∣+∣𝑤∣} ^∣𝑤∣ )

∣𝑤∣

∣𝑧∣+∣𝑤∣

= ∣𝑤∣ ⋅ 𝜓(𝑡 0 ) 𝑡 0

< ∣𝑤∣ ⋅ 𝜓( _{∣𝑢∣+∣𝑤∣} ^∣𝑤∣ )

∣𝑤∣

∣𝑢∣+∣𝑤∣

= ∥(𝑢, 𝑤)∥ ^𝜓 .

If _{∣𝑢∣+∣𝑤∣} ^∣𝑤∣ ≥ 𝑡 ⁰ , then we clearly have ∥(𝑧, 𝑤)∥ ^𝜓 = ∥(𝑢, 𝑤)∥ ^𝜓 and thus we have (ii).

(iii) ⇒(i). Assume (iii). Let 𝑡 with 0 < 𝑡 < 𝑡 ⁰ . By _{1 −𝑡+𝑡} ^𝑡 < 𝑡 0 , we have from the assumption, 𝜓(𝑡) = ∥(1 − 𝑡, 𝑡)∥ ^𝜓 > ∥(0, 𝑡)∥ ^𝜓 = 𝑡. For 𝑡 0 ≤ 𝑡 ≤ 1, we similarly have

𝜓(𝑡) = 𝑡, and thus we get (i). _■

We similarly have the following:

Theorem 2.4 Let 𝜓 ∈ Ψ ² and let 𝑡 ^′ ₀ with 0 ≤ 𝑡 ^′ 0 ≤ 1/2. Then the following are equivalent:

(i) 𝜓(𝑡) > 1 − 𝑡 for all 𝑡 with 𝑡 ^′ 0 < 𝑡 < 1, and 𝜓(𝑡) = 1 − 𝑡 for all 𝑡 with 0 ≤ 𝑡 ≤ 𝑡 ^′ 0 . (ii) 𝜓(𝑡)/(1 − 𝑡) is strictly increasing on (𝑡 ^′ 0 , 1), and 𝜓(𝑡)/(1 − 𝑡) = 1 for all 𝑡 with 0 ≤ 𝑡 ≤ 𝑡 ^′ 0 .

(iii) Let ∣𝑧∣ < ∣𝑢∣. If _{∣𝑢∣+∣𝑤∣} ^∣𝑢∣ > 𝑡 ^′ ₀ , then ∥(𝑤, 𝑧)∥ ^𝜓 < ∥(𝑤, 𝑢)∥ ^𝜓 , and if _{∣𝑢∣+∣𝑤∣} ^∣𝑢∣ ≤ 𝑡 ^′ 0 ,

then ∥(𝑤, 𝑧)∥ ^𝜓 = ∥(𝑤, 𝑢)∥ ^𝜓 .

3. New constant.

Definition 3.1 For a Banach space 𝑋 and 𝜓 ∈ Ψ ² , the function 𝛾 𝑋,𝜓 on [0, 1] is defined by

𝛾 𝑋,𝜓 (𝑡) = sup {

∥(𝑥 + 𝑡𝑦, 𝑥 − 𝑡𝑦)∥ ^𝜓 : 𝑥, 𝑦 ∈ 𝑆 ^𝑋 } .

Note that in some special cases 𝛾 𝑋,𝜓 is expressed by means of the modulus of smoothness 𝜌 𝑋 and the constant 𝛾 𝑋 . Indeed, in the case 𝜓 = 𝜓 1 (𝜓 1 is the corresponding function with ℓ 1 -norm) we have

𝛾 𝑋,𝜓

(𝑡) = 2(𝜌 𝑋 (𝑡) + 1)

and in the case 𝜓 = 𝜓 2 (𝜓 2 is the corresponding function with ℓ 2 -norm) we have 𝛾 𝑋,𝜓

(𝑡) = √

2𝛾 𝑋 (𝑡).

Proposition 3.2 For any Banach space 𝑋 and 𝜓 ∈ Ψ ² , 2𝜓( ^1−𝑡 ₂ ) ≤ 𝛾 ^𝑋,𝜓 (𝑡) ≤ 2(1 + 𝑡)𝜓( ¹ 2 ) ∀𝑡 ∈ [0, 1].

Proof Since

∥(𝑥 + 𝑡𝑥, 𝑥 − 𝑡𝑥)∥ ^𝜓 = 2𝜓( ^{1 −𝑡} ₂ ) for 𝑥 ∈ 𝑆 ^𝑋 and 𝑡 ∈ [0, 1], we have the first inequality.

We show the second inequality. Let 𝑥, 𝑦 ∈ 𝑆 ^𝑋 and 0 ≤ 𝑡 ≤ 1. From the monotonicity of ∥ ⋅ ∥ ^𝜓 ,

∥(∥𝑥 + 𝑡𝑦∥, ∥𝑥 − 𝑡𝑦∥)∥ 𝜓 ≤ ∥(1 + 𝑡, 1 + 𝑡)∥ 𝜓

= 2(1 + 𝑡)𝜓( ¹ ₂ ).

Thus the second inequality holds. _■

Proposition 3.3 For a Banach space 𝑋 and 𝜓 ∈ Ψ ² , 𝛾 𝑋,𝜓 (𝑡) = sup {

∥(𝑥 + 𝑡𝑦, 𝑥 − 𝑡𝑦)∥ ^𝜓 : 𝑥, 𝑦 ∈ 𝐵 ^𝑋 } , (1)

where 𝐵 𝑋 is the unit ball of 𝑋.

Proof Let 𝑥, 𝑦 ∈ 𝐵 ^𝑋 . Take 𝑢, 𝑣 ∈ 𝑆 ^𝑋 such that 𝑥 = ∥𝑥∥𝑢 and 𝑦 = ∥𝑦∥𝑣. Then

∥(𝑥 + 𝑡𝑦, 𝑥 − 𝑡𝑦)∥ 𝜓 = ∥(∥𝑥∥𝑢 + 𝑡𝑦, ∥𝑥∥𝑢 − 𝑡𝑦)∥ 𝜓

≤ ∥𝑥∥ + 1

2 ∥(𝑢 + 𝑡𝑦, 𝑢 − 𝑡𝑦)∥ 𝜓 + 1 − ∥𝑥∥

2 ∥(−𝑢 + 𝑡𝑦, −𝑢 − 𝑡𝑦)∥ 𝜓

≤ max{∥(𝑢 + 𝑡𝑦, 𝑢 − 𝑡𝑦)∥ 𝜓 , ∥(−𝑢 + 𝑡𝑦, −𝑢 − 𝑡𝑦)∥ 𝜓 }.

We similarly have

∥(±𝑢 + 𝑡𝑦, ±𝑢 − 𝑡𝑦)∥ ^𝜓 ≤ max{∥(±𝑢 + 𝑡𝑣, ±𝑢 − 𝑡𝑣)∥ ^𝜓 , ∥(±𝑢 − 𝑡𝑣, ±𝑢 + 𝑡𝑣)∥ ^𝜓 }.

Therefore we have ∥(𝑥+𝑡𝑦, 𝑥−𝑡𝑦)∥ ^𝜓 ≤ ∥(𝑥 ⁰ + 𝑡𝑦 0 , 𝑥 0 −𝑡𝑦 ⁰ ) ∥ ^𝜓 for some 𝑥 0 , 𝑦 0 ∈ 𝑆 ^𝑋 ,

which implies (1). This completes the proof. _■

4. Geometrical properties. We first consider the uniform non-squareness.

Theorem 4.1 Let 𝜓 ∈ Ψ ² with 𝜓 ∕= 𝜓 ∞ . Then the following are equivalent:

(i) 𝑋 is uniformly non-square.

(ii) 𝛾 𝑋,𝜓 (𝑡) < 2(1 + 𝑡)𝜓( ¹ ₂ ) for any 𝑡 with 0 < 𝑡 ≤ 1.

(iii) 𝛾 𝑋,𝜓 (𝑡 0 ) < 2(1 + 𝑡 0 )𝜓( ¹ ₂ ) for some 𝑡 0 with 0 < 𝑡 0 ≤ 1.

Proof (i) ⇒(ii): Let 𝑡 with 0 < 𝑡 ≤ 1. Assume that 𝑋 is uniformly non-square.

Then there exists a 𝛿 > 0 such that ∥𝑥 + 𝑦∥ ≤ 2(1 − 𝛿) or ∥𝑥 − 𝑦∥ ≤ 2(1 − 𝛿) for any 𝑥, 𝑦 ∈ 𝑆 ^𝑋 . If ∥𝑥 + 𝑦∥ ≤ 2(1 − 𝛿), then

∥𝑥 + 𝑡𝑦∥ = ∥𝑡(𝑥 + 𝑦) + (1 − 𝑡)𝑥∥

≤ 𝑡∥𝑥 + 𝑦∥ + (1 − 𝑡)∥𝑥∥

≤ 1 + 𝑡 − 2𝑡𝛿.

So we have

∥(𝑥 + 𝑡𝑦, 𝑥 − 𝑡𝑦)∥ ^𝜓 ≤ ∥(1 + 𝑡 − 2𝑡𝛿, 1 + 𝑡)∥ ^𝜓 .

By 𝜓 ∕= 𝜓 ∞ , 𝜓(1/2) > 1/2. Hence there exists 𝑡 0 with 1/2 < 𝑡 0 ≤ 1 such that 𝜓(𝑡) > 𝑡 for all 𝑡 with 0 < 𝑡 < 𝑡 0 , and 𝜓(𝑡) = 𝑡 for all 𝑡 with 𝑡 0 ≤ 𝑡 ≤ 1. Applying Theorem 2.3 we obtain that for 𝑧, 𝑤 with ∣𝑧∣ < ∣𝑢∣ and _{∣𝑢∣+∣𝑤∣} ^∣𝑤∣ < 𝑡 0 , we have

∥(𝑧, 𝑤)∥ ^𝜓 < ∥(𝑢, 𝑣)∥ ^𝜓 . By

1 + 𝑡

1 + 𝑡 + 1 + 𝑡 = 1 2 < 𝑡 0 , we have

∥(1 + 𝑡 − 2𝑡𝛿, 1 + 𝑡)∥ ^𝜓 < ∥(1 + 𝑡, 1 + 𝑡)∥ ^𝜓 . If ∥𝑥 − 𝑦∥ ≤ 2(1 − 𝛿), we similarly have

∥(𝑥 + 𝑡𝑦, 𝑥 − 𝑡𝑦)∥ ^𝜓 ≤ ∥(1 + 𝑡, 1 + 𝑡 − 2𝑡𝛿)∥ ^𝜓 . Applying Theorem 2.4 we get

∥(1 + 𝑡, 1 + 𝑡 − 2𝑡𝛿)∥ ^𝜓 < ∥(1 + 𝑡, 1 + 𝑡)∥ ^𝜓 . Thus

𝛾 𝑋,𝜓 (𝑡) ≤ max{∥(1 + 𝑡 − 2𝑡𝛿, 1 + 𝑡)∥ ^𝜓 , ∥(1 + 𝑡, 1 + 𝑡 − 2𝑡𝛿)∥ ^𝜓 }

< ∥(1 + 𝑡, 1 + 𝑡)∥ 𝜓 . Thus we have (ii).

(ii) ⇒ (iii): Obvious.

(iii) ⇒ (i): Suppose that 𝛾 ^𝑋,𝜓 (𝑡 0 ) < 2(1 + 𝑡 0 )𝜓( ¹ ₂ ) for some 𝑡 0 with 0 < 𝑡 0 ≤ 1. If 𝑋 is not uniformly non-square, then there exist {𝑥 ^𝑛 }, {𝑦 ^𝑛 } in 𝑆 ^𝑋 with ∥𝑥 ^𝑛 ± 𝑦 ^𝑛 ∥ → 2.

From the convexity of ∥ ⋅ ∥, we have 1

1 + 𝑡 0

𝑥 𝑛 ± 𝑡 0

1 + 𝑡 0

𝑦 𝑛

→ 1,

which imples ∥(𝑥 ^𝑛 + 𝑡 0 𝑦 𝑛 , 𝑥 𝑛 − 𝑡 ⁰ 𝑦 𝑛 ) ∥ ^𝜓 → ∥(1 + 𝑡 ⁰ , 1 + 𝑡 0 ) ∥ ^𝜓 . Thus we have 𝛾 𝑋,𝜓 (𝑡 0 ) = 2(1 + 𝑡 0 )𝜓( ¹ ₂ ).

This is a contradiction and so completes the proof. _■ Since 𝛾 𝑋,𝜓

(𝑡) = √

2𝛾 𝑋 (𝑡), we have

Corollary 4.2 ([7, 18]) Let 𝑋 be a Banach space. Then the following are equ- ivalent:

(i) 𝑋 is uniformly non-square.

(ii) 𝜌 𝑋 (𝑡) < 𝑡 for any (resp. some) 𝑡 with 0 < 𝑡 ≤ 1.

(iii) 𝛾 𝑋 (𝑡) < (1 + 𝑡) ² for any (resp. some) 𝑡 with 0 < 𝑡 ≤ 1.

Remark 4.3 In Theorem 4.1 we need the assumption that 𝜓 ∕= 𝜓 ∞ . We consider the case 𝜓 = 𝜓 _∞ . Then for any Banach space 𝑋

𝛾 𝑋,𝜓

(𝑡) = 2(1 + 𝑡)𝜓 _∞ ( ¹ ₂ ).

Indeed, it is obvious from the equality 2𝜓 _∞ ( ^{1 −𝑡} ₂ ) = 2(1 + 𝑡)𝜓 _∞ ( ¹ ₂ ) and Proposition 3.2.

For 1/2 < 𝜆 ≤ 1 we consider

𝜑 𝜆 (𝑡) =

{ 2(𝜆 − 1)𝑡 + 1 if 0 ≤ 𝑡 ≤ 1/2, 2(𝜆 − 1)(1 − 𝑡) + 1 if 1/2 ≤ 𝑡 ≤ 1.

It is obvious that 𝜑 𝜆 ∈ Ψ ² . Applying Theorem 4.1 we have the following.

Corollary 4.4 Let 𝑋 be a Banach space. Then the following are equivalent.

(i) 𝑋 is uniformly non-square.

(ii) 𝛾 𝑋,𝜑

(𝑡) < 2𝜆(1 + 𝑡) for any (resp. some) 𝑡 with 0 < 𝑡 ≤ 1.

Moreover we shall estimate 𝛾 𝑋,𝜓 for the case 𝑋 is a Hilbert space or an ℓ 𝑝 space.

Theorem 4.5 Let 𝑋 be a Hilbert space and 𝜓 ∈ Ψ ² . Assume that 𝜓 ≥ 𝜓 ² and 𝜓/𝜓 2 has the maximum at 𝑡 = ¹ ₂ . Then

𝛾 𝑋,𝜓 (𝑡) = 2(1 + 𝑡 ² ) ^1/2 𝜓( ¹ ₂ ) (2)

for all 𝑡 with 0 ≤ 𝑡 ≤ 1.

Proof Let 𝑥, 𝑦 ∈ 𝑆 ^𝑋 and 0 ≤ 𝑡 ≤ 1. Put 𝑀 ¹ = max {𝜓(𝑡)/𝜓 ² (𝑡) : 0 ≤ 𝑡 ≤ 1}.

From the assumption we have 𝑀 1 = √

2𝜓( ¹ ₂ ). Hence

∥(𝑥 + 𝑡𝑦, 𝑥 − 𝑡𝑦)∥ 𝜓 ≤ 𝑀 ¹ ∥(𝑥 + 𝑡𝑦, 𝑥 − 𝑡𝑦)∥ ^𝜓

= 𝑀 1 ( ∥𝑥 + 𝑡𝑦∥ ² + ∥𝑥 − 𝑡𝑦∥ ² ) ^1/2

= 2(1 + 𝑡 ² ) ^1/2 𝜓( ¹ ₂ ).

Therefore

𝛾 𝑋,𝜓 (𝑡) ≤ 2(1 + 𝑡 ² ) ^1/2 𝜓( ¹ ₂ ).

Take 𝑥 0 , 𝑦 0 ∈ 𝑆 ^𝑋 with ⟨𝑥 ⁰ , 𝑦 0 ⟩ = 0. Obviously,

∥(𝑥 ⁰ + 𝑡𝑦 0 , 𝑥 0 − 𝑡𝑦 ⁰ ) ∥ ^𝜓 = 2(1 + 𝑡 ² ) ^1/2 𝜓( ¹ ₂ ).

Therefore we obtain (2). _■

Corollary 4.6 ([7, 18]) Let 𝑋 be a Hilbert space. Then 𝛾 𝑋 (𝑡) = 1 + 𝑡 ² and 𝜌 𝑋 (𝑡) = √

1 + 𝑡 ² − 1 for all 𝑡 with 0 ≤ 𝑡 ≤ 1.

Theorem 4.7 Let 2 ≤ 𝑝 < ∞ and 𝜓 ∈ Ψ ² . Assume that 𝜓 ≥ 𝜓 ^𝑝 and 𝜓/𝜓 𝑝 has the maximum at 𝑡 = ¹ ₂ . Then

𝛾 ℓ

,𝜓 (𝑡) = 2 ^{1 −}

((1 + 𝑡) ^𝑝 + (1 − 𝑡) ^𝑝 ) ^1/𝑝 𝜓( ¹ ₂ ).

(3)

Proof To prove this we use the following Hanner inequality (see [18]).

∥𝑥 + 𝑦∥ ^𝑝 𝑝 + ∥𝑥 − 𝑦∥ ^𝑝 _𝑝 ≤ (∥𝑥∥ ^𝑝 + ∥𝑦∥ ^𝑝 ) ^𝑝 + ∣∥𝑥∥ ^𝑝 − ∥𝑦∥ ^𝑝 ∣ ^𝑝 𝑥, 𝑦 ∈ ℓ ^𝑝 . (4)

Let 𝑥, 𝑦 ∈ 𝑆 ^𝑋 and 0 ≤ 𝑡 ≤ 1. Put 𝑀 ² = max {𝜓(𝑡)/𝜓 ^𝑝 (𝑡) : 0 ≤ 𝑡 ≤ 1}. From the assumption we have 𝑀 2 = 2 ¹⁻

𝜓( ¹ ₂ ). By (4),

∥(𝑥 + 𝑡𝑦, 𝑥 − 𝑡𝑦)∥ 𝜓 ≤ 𝑀 ² ∥(𝑥 + 𝑡𝑦, 𝑥 − 𝑡𝑦)∥ ^𝑝

= 𝑀 2 ( ∥𝑥 + 𝑡𝑦∥ ^𝑝 𝑝 + ∥𝑥 − 𝑡𝑦∥ ^𝑝 𝑝 ) ^1/𝑝

≤ 2 ^{1 −}

((1 + 𝑡) ^𝑝 + (1 − 𝑡) ^𝑝 ) ^1/𝑝 𝜓( ¹ ₂ ).

Hence 𝛾 ℓ

,𝜓 (𝑡) ≤ 2 ¹⁻

((1 + 𝑡) ^𝑝 + (1 − 𝑡) ^𝑝 ) ^1/𝑝 𝜓( ¹ ₂ ). Put 𝑥 0 = 2 ^−1/𝑝 (1, 1, 0, ⋅ ⋅ ⋅ ), 𝑦 ⁰ = 2 ^−1/𝑝 (1, −1, 0, ⋅ ⋅ ⋅ ). Then

∥(𝑥 ⁰ + 𝑡𝑦 0 , 𝑥 0 − 𝑡𝑦 ⁰ ) ∥ ^𝜓 = 2 ^{1 −}

((1 + 𝑡) ^𝑝 + (1 − 𝑡) ^𝑝 ) ^1/𝑝 𝜓( ¹ ₂ ).

Therefore we obtain (3). _■

5. Uniform normal structure. A Banach space 𝑋 is said to have normal structure (resp. weak normal structure) if 𝑟(𝐴) < diam 𝐴 for every bounded closed convex subset (resp. weakly compact convex subset) 𝐴 of 𝑋 with diam 𝐴 > 0, where

𝑟(𝐴) = inf {sup{∥𝑥 − 𝑦∥ : 𝑦 ∈ 𝐴} : 𝑥 ∈ 𝐴}

and

diam 𝐴 = sup {∥𝑥 − 𝑦∥ : 𝑥, 𝑦 ∈ 𝐴}.

Then 𝑋 is said to have uniform normal structure if

inf {diam 𝐴/𝑟(𝐴)} > 1,

where the infimum is taken over all bounded closed convex subsets 𝐴 of 𝑋 with diam 𝐴 > 0. It is well-known that uniformly convex (resp. uniformly smooth) Banach spaces have uniform normal structure. Moreover, normal structure and uniform normal structure are important properties to obtain fixed points of non- expansive mappings (see [8, 18]).

In this section, we characterize Banach spaces with uniform normal structure in terms of 𝛾 𝑋,𝜓 .

We say that a Banach space 𝑌 is finitely representable in 𝑋 if, for any 𝜆 > 1 and each finite-dimensional subspace 𝑌 1 of 𝑌 , there is an isomorphism 𝑇 of 𝑌 1 into 𝑋 for which

𝜆 ⁻¹ ∥𝑥∥ ≤ ∥𝑇 𝑥∥ ≤ 𝜆∥𝑥∥ (∀𝑥 ∈ 𝑌 ¹ ).

Theorem 5.1 Let 𝑋, 𝑌 be Banach spaces and 𝜓 ∈ Ψ ² . If 𝑌 is finitely representable in 𝑋, then 𝛾 𝑋,𝜓 (𝑡) ≥ 𝛾 ^𝑌,𝜓 (𝑡) for any 𝑡 with 0 ≤ 𝑡 ≤ 1.

Proof Assume that 𝑌 is finitely representable in 𝑋. Fix 𝑡 with 0 ≤ 𝑡 ≤ 1, and let 𝑥, 𝑦 ∈ 𝑆 ^𝑌 . We put 𝑌 1 = span {𝑥, 𝑦}. From the assumption, for 𝜆 > 1, we can find an isomorphism 𝑇 from 𝑌 1 into 𝑋 such that

𝜆 ⁻¹ ∥𝑧∥ ≤ ∥𝑇 𝑧∥ ≤ 𝜆∥𝑧∥, ∀𝑧 ∈ 𝑌 ¹ . Since 𝜆 ⁻¹ 𝑇 𝑥, 𝜆 ⁻¹ 𝑇 𝑦 ∈ 𝐵 ^𝑋 , we have from Proposition 3.3,

𝛾 𝑋,𝜓 (𝑡) ≥ ∥(𝜆 ⁻¹ 𝑇 𝑥 + 𝑡𝜆 ⁻¹ 𝑇 𝑦, 𝜆 ⁻¹ 𝑇 𝑥 − 𝑡𝜆 ⁻¹ 𝑇 𝑦) ∥ 𝜓

= 𝜆 ⁻¹ ∥(𝑇 (𝑥 + 𝑡𝑦), 𝑇 (𝑥 − 𝑡𝑦))∥ ^𝜓

≥ 𝜆 ⁻² ∥(𝑥 + 𝑡𝑦, 𝑥 − 𝑡𝑦)∥ ^𝜓 ,

which implies 𝛾 𝑋,𝜓 (𝑡) ≥ 𝜆 ⁻² 𝛾 𝑌,𝜓 (𝑡). As 𝜆 → 1, we obtain 𝛾 ^𝑋,𝜓 (𝑡) ≥ 𝛾 ^𝑌,𝜓 (𝑡). _■ We first consider Banach spaces having normal structure. We need the following lemma.

Lemma 5.2 ([18], Lemma 4.5) Let 𝑋 be a Banach space without weak normal structure. For any 𝜂 with 0 < 𝜂 < 1 and any 𝑡 with 0 ≤ 𝑡 ≤ 1, there exist 𝑥 ∈ 𝑆 ^𝑋 and 𝑦, 𝑧 ∈ 𝑡𝑆 ^𝑋 such that

(i) 𝑦 − 𝑧 = 𝑎𝑥, with ∣𝑎 − 𝑡∣ < 𝜂, (ii) ∥𝑥 − 𝑦∥ > 1 − 3𝜂,

(iii) ∥𝑥 + 𝑦∥ > 1 + 𝑡 − 3𝜂, ∥𝑥 − 𝑧∥ > 1 + 𝑡 − 3𝜂.

Theorem 5.3 Let 𝑋 be a Banach space and let 𝜓 ∈ Ψ ² with 𝜓 ∕= 𝜓 ∞ . If there

exists 𝑡 with 0 < 𝑡 ≤ 1 such that 𝛾 ^𝑋,𝜓 (𝑡) < (2 + 𝑡)𝜓( _2+𝑡 ¹ ), then 𝑋 has normal

structure.

Proof Assume that 𝑋 does not have weak normal structure. From Lemma 5.2, for any 𝜂 with 0 < 𝜂 < 1 and 𝑡 with 0 < 𝑡 ≤ 1, there exist 𝑥 ∈ 𝑆 ^𝑋 , 𝑦 ∈ 𝑡𝑆 ^𝑋 such that ∥𝑥 + 𝑦∥ > 1 + 𝑡 − 3𝜂 and ∥𝑥 − 𝑦∥ > 1 − 3𝜂. Hence

𝛾 𝑋,𝜓 (𝑡) ≥ ∥(𝑥 + 𝑦, 𝑥 − 𝑦)∥ 𝜓

≥ ∥(1 + 𝑡 − 3𝜂, 1 − 3𝜂)∥ 𝜓 . Since 𝜂 is arbitrary, we have

𝛾 𝑋,𝜓 (𝑡) ≥ ∥(1 + 𝑡, 1)∥ 𝜓

= (2 + 𝑡)𝜓( _2+𝑡 ¹ ).

This is a contradiction. Thus 𝑋 has weak normal structure. Also, since 𝛾 𝑋,𝜓 (𝑡) < (2 + 𝑡)𝜓( _2+𝑡 ¹ )

= ∥(1 + 𝑡, 1)∥ 𝜓

≤ ∥(1 + 𝑡, 1 + 𝑡)∥ 𝜓

= 2(1 + 𝑡)𝜓( ¹ ₂ ),

it follows that 𝑋 is uniformly non-square and hence is reflexive. Thus 𝑋 has normal structure (see Theorem 4.7 in [18]). This completes the proof. _■ We say that a Banach space 𝑋 has super-normal structure if every Banach space which is finitely representable in 𝑋 has normal structure. In order to characterize Banach spaces with uniform normal structure, we need the following lemma.

Lemma 5.4 ([12]) Every super-normal structure Banach space has uniform normal structure.

Theorem 5.5 Let 𝑋 be a Banach space and let 𝜓 ∈ Ψ ² with 𝜓 ∕= 𝜓 ∞ . If there exists 𝑡 with 0 < 𝑡 ≤ 1 such that

𝛾 𝑋,𝜓 (𝑡) < (2 + 𝑡)𝜓( _2+𝑡 ¹ ), then 𝑋 has uniform normal structure.

Proof Let 𝑌 be a Banach space which is finitely representable in 𝑋. By Lemma 5.4, it is enough to show that 𝑌 has normal structure. By Theorem 5.1, we have

𝛾 𝑌,𝜓 (𝑡) ≤ 𝛾 ^𝑋,𝜓 (𝑡) < (2 + 𝑡)𝜓( _2+𝑡 ¹ )

for 𝑡 with 0 < 𝑡 ≤ 1. Thus it follows from Theorem 5.3 that 𝑌 has normal structure.

This completes the proof. _■

Corollary 5.6 ([7, 18]) Let 𝑋 be a Banach space. If there exists 𝑡 with 0 < 𝑡 ≤ 1

such that 𝛾 𝑋 (𝑡) < ^1+(1+𝑡) ₂

(resp. 𝜌 𝑋 (𝑡) < ₂ ^𝑡 ), then 𝑋 has uniform normal structure.

Corollary 5.7 Let 𝑋 be a Banach space. If there exists 𝑡 with 0 < 𝑡 ≤ 1 such that

𝛾 𝑋,𝜑

(𝑡) < 2𝜆 + 𝑡 2 + 𝑡 , then 𝑋 has uniform normal structure.

References

[1] J. Banas and B. Rzepka, Functions related to convexity and smoothness of normed spaces, Rend. Circ. Mat. Palermo 46(2) (1997), no. 3, 395–424.

[2] B. Beauzamy, Introduction to Banach Spaces and Their Geometry, 2nd ed., North-Holland, Amsterdam-New York-Oxford, 1985.

[3] F. F. Bonsall and J. Duncan, Numerical Ranges II, London Math. Soc. Lecture Note Series, Vol.10, 1973.

[4] M. M. Day, Normed Linear Spaces, 3rd ed., in: Ergebnisse der Mathematik und ihrer Gren- zgebiete, Vol. 21, Springer-Verlag, New York, 1973.

[5] S. Dhompongsa, P. Piraisangjun and S. Saejung, Generalized Jordan-von Neumann constants and uniform normal structure, Bull. Austral. Math. Soc. 67 (2003), 225–240.

[6] P. N. Dowling, On convexity properties of 𝜓-direct sums of Banach spaces, J. Math. Anal.

Appl. 288 (2003), no. 2, 540–543.

[7] J. Gao, Normal structure and modulus of smoothness in Banach spaces, Nonlinear Funct.

Anal. Appl. 8 (2003), no. 2, 233–241.

[8] J. Gao and K. S. Lau, On two classes of Banach spaces with uniform normal structure, Studia Math. 99 (1991), 41–56.

[9] M. Kato, K. -S. Saito and T. Tamura, On 𝜓-direct sums of Banach spaces and convexity, J.

Austral. Math. Soc. 75 (2003), 413-422.

[10] M. Kato, K. -S. Saito and T. Tamura, Uniform non-squareness of 𝜓-direct sums of Banach spaces 𝑋 ⊕

𝑌 , Math. Inequalities Appl. 7 (2004), 429-437.

[11] M. Kato, L. Maligranda and Y. Takahashi, On James and Jordan-von Neumann constants and the normal structure coefficient of Banach spaces, Studia Math. 144 (2001), 275–295.

[12] M. A. Khamsi, Uniform smoothness implies super-normal structure property, Nonlinear Anal.

19 (1992) 1063-1069.

[13] R. E. Megginson, An Introduction to Banach Space Theory, Grad. Texts in Math. 183, Springer, New York, 1998.

[14] K.-I. Mitani and K.-S. Saito, A note on geometrical properties of Banach spaces using 𝜓- direct sums, J. Math. Anal. Appl. 327 (2007), 898–907.

[15] K.-S. Saito and M. Kato, Uniform convexity of 𝜓-direct sums of Banach spaces, J. Math.

Anal. Appl. 277 (2003), 1–11.

[16] Y. Takahashi and M. Kato, Von Neumann-Jordan constant and uniformly non-square Ba-

nach spaces, Nihonkai Math. J. 9 (1998), 155–169.

[17] Y. Takahashi, M. Kato and K.-S. Saito, Strict convexity of absolute normes on ℂ

and direct sums of Banach spaces, J. Inequal. Appl. 7 (2002), 179–186.

[18] C. Yang and F. Wang, On a new geometric constant related to the von Neumann-Jordan constant, J. Math. Anal. Appl. 324 (2006), 555–565.

Ken-Ichi Mitani

Department of Applied Chemistry and Biotechnology, Faculty of Engineering, Niigata Institute of Technology

Kashiwazaki, Niigata 945-1195, Japan E-mail: mitani@adm.niit.ac.jp Kichi-Suke Saito

Department of Mathematics, Faculty of Science, Niigata University Niigata 950-2181, Japan

E-mail: saito@math.sc.niigata-u.ac.jp

(Received: 22.05.2008)