Article Search
eISSN 0454-8124
pISSN 1225-6951

### Article

Kyungpook Mathematical Journal 2023; 63(1): 61-78

Published online March 31, 2023

### Some Geometric Constants Related to the Heights and Midlines of Triangles in Banach Spaces

Dandan Du, Yuankang Fu, Zhijian Yang and Yongjin Li*

Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, P.R. China
e-mail : dudd5@mail2.sysu.edu.cn, fuyk5@mail2.sysu.edu.cn, yangzhj55@mail2.sysu.edu.cn and stslyj@mail.sysu.edu.cn

Received: April 19, 2022; Revised: October 6, 2022; Accepted: October 7, 2022

In this paper, we introduce two new geometric constants related to the heights of triangles: ΔH(X) and Δh(X,I). We also propose two new geometric constants, Δm(X) and Δm(X), related to the midlines of equilateral triangles, and discuss the relation between the heights and midlines in equilateral triangles. We give estimates for these geometric constants in terms of other geometric parameters, and the geometric constants are used to discuss geometric properties such as uniform non-squareness, uniform normal structure, and the fixed point property.

Keywords: uniform non-squareness, uniform normal structure, fixed point property

Geometric constants has been widely studied, as they make it easier to deal with certain problems in Banach space. They do this because they not only essentially reflect the geometric properties of a space X, but also enables us to study the geometric properties of Banach spaces quantitatively. For example, the modulus of convexity introduced by Clarkson [7] can be used to characterize uniformly convex spaces, the modulus of smoothness proposed by Day [9] can be used to characterize uniformly smooth spaces. We refer the readers to the papers [2-4, 13-15, 22, 28] about the modulus of convexity and the modulus of smoothness.

It is well known that the height and midline of a triangle play an important role in Euclidean geometry, as does the geometric theory of Banach spaces. In 1997, Alsina et al. obtained a collection of new characterizations of inner product spaces by using well-known formulas about the height of a triangle in [1]. In 2009, Ni et al. [27] proposed a new geometric constant h(X) related to the heights of equilateral triangles, which can be used to characterize the uniformly non-square spaces. In Functional Analysis, various fine geometric properties of finite dimensional spaces (i.e. Minkowski spaces) play important roles in the so-called local theory of Banach spaces (see [24, 30]). In 2001, Martini et al. [23] obtained the upper bounds of medians for equilateral triangles in any Minkowski planes (i.e. two-dimensional Minkowski spaces).

Inspired by the excellent works mentioned above, in this paper, we will introduce some geometric constants related to the heights and midlines of triangles in non-trivial Banach spaces, and use them to study some properties of Banach spaces. This paper is organized in the following way:

In Section 2, we recall some fundamental concepts and conclusions that we need to use in subsequent discussions.

In Section 3, we consider the following three constants:

Δh(X)=infinfλλx+(1λ)y:x=y=1,xy=1, ΔH(X)=supinfλλx+(1λ)y:x=y=1,xy=1, Δh(X,I)=infinfλλx+(1λ)y:x=y=1,xIy.

The first one is the constant h(X) proposed by Ni et al. [27], and the latter two are proposed by us. These three constants are closely related to the height of the triangle. The two constants Δh(X) and ΔH(X) are related to the heights of equilateral triangles, and the constant Δh(X,I) is related to the heights on the hypotenuses of right triangles. The bounds of these constants and the values of these constants for some specific spaces will be given. Further, some estimates of these constants in terms of other constants will also be discussed. In particular, the relationships between these constants and some geometric properties of Banach spaces will be studied, including uniform non-squareness, uniform convexity, strict convexity and uniform normal structure.

In Section 4, we introduce the following third constants:

Δm(X)=infx+y2:x=y=1,xy=1, ΔM(X)=supx+y2:x=y=1,xy=1, DΔ(X)=supx+yinfλ[0,1]λx+(1λ)y:x=y=xy=1.

The first two constants are both related to the midlines of equilateral triangles. The bounds of these two constants and the values of these constants for some specific spaces will be given. Further, a necessary and sufficient condition and a sufficient condition for uniformly non-square space will be established. Finally, we introduce a new constant DΔ(X) related to the difference between the midilines and heights in equilateral triangles.

Throught the paper, let X be a non-trivial Banach space, that is, dimX2, and using SX and BX to represent the unit sphere and closed unit ball of X, respectively.

Recall that the Banach space X is called uniformly non-square [17], if there exists a constant δ(0,1) such that for any x,ySX, either x+y21δ or xy21δ.

The constants

J(X)=sup{min{x+y,xy}:x,ySX} S(X)=inf{max{x+y,xy}:x,ySX}

are defined by Gao [11] in order to measure the degree of uniform non-squareness, and usually called the James constant and Schäffer constant, respectively.

Later, the equivalent definitions of James constant and Schäffer constant

J(X)=sup{x+y:x,ySX,x+y=xy} S(X)=inf{x+y:x,ySX,x+y=xy}

are introduced by He and Cui [16].

Now let us collect some properties of the two constants for non-trivial Banach spaces (see [16, 20, 21]):

(1) 1S(X)2J(X)2.

(2) J(X)S(X)=2.

(3) Let X be a Banach space with dimX3. Then J(X)=2 if and only if X is a Hilbert space.

(4) X is uniformly non-square if and only if either J(X)<2 or S(X)>1.

Recall that the Banach space X is called uniformly convex [7], if, for any ϵ>0, there exists δ>0, such that for any x,ySX with xy>ϵ, then x+y2<1δ.

The Clarkson modulus of convexity of a Banach space X

δX(ϵ)=inf1x+y2:x,ySX,xy=ϵ,(0ϵ2)

was proposed in [7] and can be used to characterize uniformly convex space.

Some famous conclusions about δX(ϵ) are listed in [13, 28, 22]:

(1) For all Banach spaces X, then

δX(ϵ)11ϵ24.

(2) For any x,yX such that x2+y2=2, then

x+y244δXxy2.

(3) X is uniformly non-square if and only if there exists ϵ(0,2) such that δX(ϵ)>0.

In order to get a better understanding of some geometric properties of Banach spaces, in [14] Gurariy introduced the βX modulus of as the function

βX(ϵ)=inf1infa[0,1]ax+(1a)y:x,ySX,xy=ϵ,(0ϵ2)

and the various properties of this constant were given in [15, 3]:

(1) For any ϵ[0,2], δX(ϵ)βX(ϵ)2δX(ϵ).

(2) Let X be a Hilbert space, then for each ϵ[0,2],

βX(ϵ)=11ϵ2412.

(3) Banach space X is uniformly convex if and only if βX(ϵ)>0, for any ϵ[0,2].

Recall that Banach space X is said to be uniformly smooth [9], whenever given 0<ϵ2, there exists δ>0 such that if xSX and yδ, then

x+y+xy<2+ϵy.

In order to study uniformly smooth space, the modulus of smoothness ρX(τ):[0,+)[0,+) was introduced by Day [9] as follows:

ρX(τ)=supx+τy+xτy21:x,ySX,(τ0).

The following function ρ(ϵ):[0,2][0,1], which we call it the modulus of smoothness and can be used to characterize uniformly smooth space, was introduced by Banaś [2] as follows:

ρ(ϵ)=sup1x+y2:x,ySX,xy=ϵ,(0ϵ2).

Later, Baronti and Papini [4] considered the following related modulus:

ρ(ϵ)=sup1λx+(1λ)y:x,ySX,xy=ϵ,λ[0,1],(0ϵ2).

Some important conclusions about ρ(ϵ) and ρ(ϵ) are listed as follows [4]:

(1) For all Banach space X, then

ρ(ϵ)11ϵ24.

(2) X is not uniformly non-square if and only if ρ(1)=12.

(3) X is not uniformly non-square if and only if ρ(1)=12.

### 3. The Heights of Triangles

In this section, we will study two types of constants. The first type is closely related to the heights of equilateral triangles, and the second type is related to the heights on the hypotenuses of right triangles.

### 3.1. The heights of equilateral triangles

Ni et al. [27] introduced the following constant

Δh(X)=infinfλλx+(1λy) : x=y=1, xy=1.

Inspired by Δh(X), we consider the following constant

ΔH(X)=supinfλλx+(1λ)y : x=y=1, xy=1.

Next, we give the equivalent forms of Δh(X) and ΔH(X), which will help us understand their geometric meanings.

Proposition 3.1. Let X be a Banach space. Then

(1) Δh(X)=infinfλ[0,1]λx+(1λ)y : x=y=1, xy=1,

(2) ΔH(X)=supinfλ[0,1]λx+(1λ)y : x=y=1, xy=1.

Proof. (1) First, for any x,ySX and λ[0,1], we can get

λx+(1λ)yλx+(1λ)y=1.

Second, for any x,ySX and λ[1,+],

we can obtain

λx+(1λ)yλx(λ1)y=1.

Further, for any x,ySX and λ[,0], we can also obtain

λx+(1λ)y||λ||1λ||=1.

Thus, we can conclude

Δh(X)=infinfλλx+(1λ)y : x=y=1, xy=1  =infinfλ[0,1]λx+(1λ)y : x=y=1, xy=1.

(2) The proof is similar to (1), and we omit it.

The geometric meanings of Δh(X) and ΔH(X) : consider the Euclidean plane with OA=x, OB=y, then we have BA=xy. Assume that OCAB, the geometric explanations of Δh(X) and ΔH(X) are the infimum and supremun of the heights on the side AB on the equilateral triangles ΔOAB, respectively.

Figure 1. Geometric explanations of Δh(X) and ΔH(X)

### 3.1.1. The bounds of Δh(X) and ΔH(X)

First, we give the bounds of Δh(X) and ΔH(X).

Proposition 3.2. Let X be a Banach space. Then

(1) 12Δh(X)32,

(2) 31ΔH(X)1.

Proof. (1) In [27], Ni et al. have obtained Δh(X)12. Moreover, it is clear that

Δh(X)infx+y2:x=y=xy=1=1ρ(1).

Then, we can obtain

Δh(X)1ρ(1)111124=32,

which completes the proof.

(2) From the definition of ΔH(X) and βX(ϵ), we can obtain ΔH(X)=1βX(1). Since βX(ϵ)2δX(ϵ) for any ϵ[0,2], we see that

ΔH(X)=1βX(1)12δX(1)1211124=31.

Note that

λx+(1λ)yλx+(1λ)y=λ+(1λ)=1,

for each λ[0,1] and x=y=xy=1, which means that ΔH(X)1. This completes the proof.

Next, we give the values of Δh(X) and ΔH(X) of some specific spaces. Example 3.3 and Example 3.4 illustrate that the lower bound of Δh(X) can be reached. Example 3.5 shows that the two constants are in general different and the upper bound of ΔH(X) can be attained. We can also obtain the exact values of Δh(X) and ΔH(X) for Hilbert spaces by Proposition 3.6.

Example 3.3. Let X=(2,1). Then Δh(X)=12.

Proof. Let x=(12,12) and y=(12,12). It is easy for us to obtain x1=y1=xy1=1. Then we can get

12Δh(X)x+y12=12.

Thus, we can obtain Δh(X)=12.

Example 3.4. Let X=(2,). Then Δh(X)=12.

Proof. Let x=(0,1) and y=(1,0). It is clear that x=y=xy=1. Thus, we can obtain

12Δh(X)x+y2=12,

which means that Δh(X)=12.

Example 3.5. Let X be the space 2 endowed with the norm

x=(x1,x2)=(x1 ,x2 ) ,(x1 x2 0),(x1 ,x2 )1 ,(x1 x2 0).

Then Δh(X)=34 and ΔH(X)=1.

Proof. From Theorem 4 in [4], we can get ρ(1)=ρ(1). Then by Example 1 in [4], we can obtain ρ(1)=14. Thus Δh(X)=1ρ(1)=34.

Moreover, from [3], βX(ϵ)=max0,11ϵ. So, we have βX(1)=0. Thus, we can obtain ΔH(X)=1βX(1)=1.

Proposition 3.6. Let X be a Hilbert space. Then Δh(X)=ΔH(X)=32.

Proof. Let x,ySX such that xy=1. Since X is a Hilbert space, then

1=xy2=x22x,y+y2=22x,y,

which means that x,y=12. Then for λ∈[0,1], we have

λx+(1λ)y2=λ2x2+2λ(1λ)x,y+(1λ)2y2=λ2+λ(1λ)+(1λ)2=λ2λ+1infλ[0,1]λ2λ+1=34.

Then, we can conclude that Δh(X)=32 from Proposition 3.2 and the above inequality.

On the other hand, since ΔH(X)=1βX(1) and X is a Hilbert space, then we can obtain

ΔH(X)=1βX(1)=11112412=32.

This completes the proof.

### 3.1.2. The estimates for ΔH(X) by other geometric constants

In this section, we will give some estimates for ΔH(X) in terms of other geometric constants. Firstly, we give the relation between ΔH(X) and the modulus of convexity δX(ε), the modulus of smoothness ρX(τ).

Proposition 3.7. Let X be a finite-dimensional Banach space and δ=2δX(1). Then

ΔH(X)1δ+δ2.

Proof. Since X is a finite-dimensional Banach space, we can choose x=y=xy=1 and x+y=2δ. Notice that if z1=z2=z1z2=1, then z1+z22δ and, in particular, 2xy2δ.

Now for λ[12,1], we have

λx+(1λ)y=λ(x+y)+(12λ)y      λx+y(2λ1)y=λ(2δ)2λ+1=1λδ,

and also

λx+(1λ)y=(2λ)x(1λ)(2xy)      (2λ)x(1λ)2xy      (2λ)(1λ)(2δ)=λ(1δ)+δ.

Thus, we can obtain

λx+(1λ)ymaxλ[12,1]{1λδ,λ(1δ)+δ},

which implies λx+(1λ)y1δ+δ2. The same result is also valid for λ[0,12]. Thus, we can get

ΔH(X)=supinfλ[0,1]{λx+(1λ)y} : x=y=1, xy=1  1δ+δ2.

This completes the proof.

Proposition 3.8. Let X be a Banach space. Then ΔH(X)21δX(12).

Proof. Let λ=12, we have

ΔH(X)supx+y2:x=y=xy=1.

Now, let x,ySX such that xy=1. Then we can obtain x2+y2=2. Thus, we can get

x+y244δX12,

which deduces the desired inequality.

Proposition 3.9. Let X be a Banach space. Then ΔH(X)ρX(1)+12.

Proof. Let λ=12, then λx+(1λ)y=x+y2. Thus, we can obtain

ΔH(X)supx+y2:x=y=xy=1.

Then, we can get

ΔH(X)12sup{x+y+xy:x=y=xy=1}12  12sup{x+y+xy:x,ySX}12  ρX(1)+112=ρX(1)+12,

which completes the proof.

In order to study the relationship between the constant ΔH(X) and the the von Neumann-Jordan constant CNJ(X), we give the definition of the constant CNJ(X) as follows (see [8]):

CNJ(X)=supx+y2+xy22(x2+y2):x,yX,(x,y)(0,0).

Proposition 3.10. Let X be a Banach space. Then ΔH(X)124CNJ(X)1}.

Proof. Let λ=12, we can get

ΔH(X)supx+y2:x=y=xy=1.

For any x,yX, we have

x+y2+xy22CNJ(X)(x2+y2).

Hence, for any x,ySX such that xy=1, we can obtain

x+y4CNJ(X)1,

then we can conclude the desired inequality by a simple calculation.

### 3.1.3. The relationships between Δh(X), ΔH(X) and some geometric properties of Banach spaces

In this section, we will discuss the relationships between Δh(X), ΔH(X) and some geometric properties of Banach spaces, including uniformly non-square, uniform convex and strictly convex.

First, the relationships between Δh(X), ΔH(X) and uniformly non-square are shown as follows.

Proposition 3.11. Let X a be Banach space. Then X is uniformly non-square if and only if Δh(X)>12.

Proof. The Corollary 3 in [4] shows that X is not uniformly non-square if and only if ρ(1)=12. Since it is easy for us to obtain Δh(X)=1ρ(1), then we can get Δh(X)=12 if and only if X is not uniformly non-square. This completes the proof.

Remark 3.12. The above conclusion has been proved in [27], but our proof is different and more concise.

Theorem 3.13. If ΔH(X)<1, then X is uniformly non-square.

Proof. If X is not uniformly non-square, then for any ϵ(0,2), we have δX(ϵ)=0. Thus, we can get δX(1)=0. Since for each ϵ(0,2), δX(ϵ)βX(ϵ)2δX(ϵ), we can obtain βX(1)=0. Then, we can get ΔH(X)=1βX(1)=1. This contradicts ΔH(X)<1, hence X is a uniformly non-square Banach space.

Some sufficient conditions for fixed point property, followed from the fact proved in [12] that uniformly non-square Banach spaces have the fixed point property, are presented in the following corollary.

Corollary 3.14. Assume X is a Banach space with Δh(X)>12 or ΔH(X)<1. Then X has the fixed point property.

Proposition 3.15. If Banach space X is uniformly convex, then ΔH(X)<1.

Proof. If X is uniformly convex, then βX(1)>0. Thus, we can get ΔH(X)=1βX(1)<1.

Recall that the Banach space X is called strictly convex, if for any x,ySX and xy, then x+y<2. Now, we give the relationship between strict convexity and ΔH(X).

Proposition 3.16. Let X be a finite-dimensional Banach space. If ΔH(X)=1, then X is not strictly convex.

Proof. Assume that ΔH(X)=1. Since the unit sphere of finite-dimensional Banach space is compact, so there exist x,ySX satisfying xy=1, such that for any λ[0,1], λx+(1λ)y=1. By Hahn-Banach Theorem, there exists f=1 satisfying f(λx+(1λ)y)=1. It is easily seen that f(x)=f(y)=1, then we have

fx+y2=12[f(x)+f(y)]=1.

Thus we can obtain x+y2=1, then X is not strictly convex.

### 3.2. The heights on the hypotenuse of right triangles

In order to introduce the new geometric constant and obtain our main results, we firstly give some basic concepts and some related geometric constants.

Birkhoff [5] introduced Birkhoff orthogonality: x is said to be Birkhoff orthogonality to y (denoted by xBy) if x+tyx, for any t.

Later, James [18] introduced isosceles orthogonality: x is said to be isosceles orthogonality to y (denoted by xIy) if x+y=xy.

In [19], Ji and Wu introduced a new geometric constant D(X) to give a quantitative characterization of the difference between Birkhoff orthogonality and isosceles orthogonality as follows:

D(X)=infinfλx+λy:x,ySX,xIy.

Moreover, Mizuguchi [25] further introduced a new geometric constant IB(X) to measure the difference between Birkhoff orthogonality and isosceles orthogonality in the entire normed space X as follows:

IB(X)=infinfλx+λyx:x,yX,x,y0,xIy.

Motivated by the two constants Δh(X) and ΔH(X) related to the heights of equilateral triangles. In this section, we consider the following geometric constant

Δh(X,I)=infinfλλx+(1λ)y:x=y=1,xIy

and use it to study some geometric properties.

Similarly, from the proof of Proposition 3.1, we can obtain the equivalent form of Δh(X,I):

Δh(X,I)=infinfλ[0,1]λx+(1λ)y:x=y=1,xIy.

The geometric meaning of Δh(X,I) is shown in Figure 2: consider the unit sphere in the Euclidean plane with AB=x and AC=y. Suppose ABAC, AE is the height of the hypotenuse BC on the right triangle ΔABC. Apparently, the Δh(X,I) is the infimum of the heights of the hypotenuses of the right triangles.

Figure 2. Geometric explanation of Δh(X,I).

Firstly, we give the following result which will help us to give the bounds of Δh(X,I).

Proposition 3.17. Let X be a Banach space. Then 22IB(X)22D(X)Δh(X,I)S(X)2.

Proof. Since IB(X)D(X), we can obtain the first inequality.

On the other hand, from Proposition 3.8 in [25], we have

D(X)=2infλx+(1λ)yx+y:x,ySX,xIy,0λ1.

Thus, for any x,ySX, xIy and 0λ1, we can get

D(X)2λx+(1λ)yx+y.

Since

J(X)=supx+y:x,ySX,xIy

and J(X)2, then for any x,ySX with xIy, we have

D(X)2λx+(1λ)y.

Then, we can get the second inequality.

Finally, let λ=12, then we can get

Δh(X,I)infx+y2:x,ySX,xIy=S(X)2.

Then, we can obtain the third inequality. This completes the proof.

Corollary 3.18. If D(X)>2(21), then J(X)<1+22.

Proof. From Proposition 3.17, if D(X)>2(21), then we can obtain S(X)>422. Applying J(X)S(X)=2, we have

J(X)=2S(X)<2422=1+22,

which completes the proof.

Remark 3.19. The Theorem 3.2 in [29] shows that if D(X)>2(21), then J(X)<2. Since 1+22<2, we improve the Theorem 3.2 in [29] by Corollary 3.18.

From Proposition 3.17, we can also obtain the following estimate.

Proposition 3.20. Let X be a Banach space. Then 24Δh(X,I)22.

Proof. From Proposition 3.17 and IB(X)12 (see Theorem 3.2 in [25]), we can obtain Δh(X,I)22IB(X)24. On the other hand, from Proposition 3.17 and S(X)2 by Remark 11 in [16], we can get Δh(X,I)22. This completes the proof.

The following conclusion will show two important things. One is the fact that the upper bound of Δh(X,I) in the Proposition 3.20 is sharp. The other is the Hilbert space can be characterized by Δh(X,I).

Theorem 3.21. Let X be a Banach space of dimX3. Then Δh(X,I)=22 if and only if X is a Hilbert space.

Proof. If X is a Hilbert space, then IB(X)=1 by Theorem 3.2 in [25]. Thus we can obtain Δh(X,I)=22 from Proposition 3.17 and S(X)2.

On the other hand, let Δh(X,I)=22. From Proposition 3.17 and S(X)2, we can obtain S(X)=2, then J(X)=2. By Theorem 2.3 in [21], we can get that X is a Hilbert space.

Now, we will give the relation between uniform non-squareness and Δh(X,I).

Theorem 3.22. Let X be a Banach space.

(1) If Δh(X,I)>12, then X is uniformly non-square.

(2) If X is uniformly non-square, then Δh(X,I)>24.

Proof. (1) Let X be not uniformly non-square, then S(X)=1 (see [16]). From Proposition 3.17, we can obtain

Δh(X,I)S(X)2=12.

This contradicts Δh(X,I)>12, hence X is a uniformly non-square Banach space.

(2) If X is uniformly non-square, then we can obtain IB(X)>12 by Corollary 3.6 in [25]. So, we can get Δh(X,I)>24 from Proposition 3.17. This completes the proof.

Corollary 3.23. Assume that X is a Banach space with Δh(X,I)>24. Then X has the fixed point property.

In the next portion, we will see that the constant Δh(X,I) and the uniform normal structure has a nice relationship. Brodskii and Milman [6] introduced some geometric concepts for the first time in 1948 as:

Definition 3.24. Let K be a non-singleton subset of a Banach space X, if K is closed, bounded as well as convex, then X holds the normal structure, whenever r(K)<diam(K) for every K, and consequently defined mathematically as is

diam(K):=sup{xy:x,yK}

and

r(K):=inf{sup{xy:yK}:XK},

where diam(K) and r(K) are respectively symbolized for diameter as well as for Chebyshev radius. A Banach space X is said to have uniform normal structure if

infdiam(K)r(K)>1,

with diam(K)>0.

In order to study the relationship between Δh(X,I) and uniform normal structure. Now, we give the corresponding lemma as follows:

Lemma 3.25. (see [10]) Let X be a Banach space with J(X)<1+52. Then X has uniform normal structure.

Theorem 3.26. Let X be a Banach space. If Δh(X,I)>512, then X has uniform normal structure.

Proof. Since J(X)S(X)=2, by using Proposition 3.17, we have

Δh(X,I)S(X)2=1J(X).

Hence, we can obtain

J(X)1Δh(X,I)<1512=5+12.

By utilizing Lemma 3.25, we can get that X has uniform normal structure.

### 4. The Difference Between the Heights and Midlines in Equilateral Triangles

In this section, we will introduce the two constants Δm(X) and ΔM(X) related to the midlines in equilateral triangles and the relationship between the heights and midlines in equilateral triangles. Meanwhile, we also introduce a new constant DΔ(X) to study the difference between the heights and midlines in equilateral triangles.

### 4.1. The midlines in equilateral triangles

In [23], Martin et al. have stated that, in any Minkowski planes,

infx+y:x=y=xy=13

and

supx+y:x=y=xy=13.

This result illustrates that there always exists the equilateral triangle with a median 32 as well as a median 32. Motivated by this conclusion, in this section, we will introduce two new constants Δm(X), ΔM(X) related to the midlines of equilateral triangles in non-trivial Banach spaces.

Definition 4.1. For a given Banach space X. Let

Δm(X)=infx+y2 : x=y=1,xy=1, ΔM(X)=supx+y2 : x=y=1,xy=1.

First, we give their bounds.

Proposition 4.2. Let X be a Banach space. Then 12Δm(X)32ΔM(X)1.

Proof. Clearly, 12Δm(X) and ΔM(X)1 can be given the following two inequalities

x+y2x+y2=1, 1=x=2x2=(x+y)+(xy)2x+y+xy2=12+x+y2,

where x,ySX with xy=1.

Then, by the definition of δX(ϵ) and ρ(ϵ), we can obtain

Δm(X)=1ρ(1)111124=32

and

ΔM(X)=1δX(1)111124=32

which is the desired conclusion.

Remark 4.3. From the proof of Example 3.3, we can obtain Δm(2,1)=Δh(2,1)=12. And we can also get ΔM(X)=ΔH(X)=1, if X is the space 2 endowed the norm as Example 3.5. Thus, the lower bound and the upper bound of Δm(X) and ΔM(X) can be attained, respectively.

With regard to the other bounds of Δm(X) and ΔM(X), Proposition 4.4 shows that they can be attained in Hilbert spaces.

Proposition 4.4. Let X be a Hilbert space, then Δm(X)=ΔM(X)=32.

Proof. Let x,ySX such that xy=1. By applying the parallelogram law, we can obtain

x+y2=2x2+2y2xy2=3,

it is easy for us to get Δm(X)=ΔM(X)=32.

Corollary 4.5. If X is a Hilbert space, then Δh(X)=Δm(X) and ΔH(X)=ΔM(X) from Proposition 3.6 and Proposition 4.4, which implies that the heights coincide with the midlines of equilateral triangles in Hilbert spaces. However, the converse is not true by Remark 4.3.

Now, we will discuss the relation between Δm(X) and the Schäffer constant S(X).

Proposition 4.6. Let X be a Banach space. Then Δm(X)S(X)2.

Proof. Let x,ySX such that ∥x-y∥=1. From the proof of Proposition 4.2, we can obtain x+y1. Thus, we can get

2Δm(X)=inf{max{x+y,xy}:x,ySX,xy=1}.

Then it is easily seen that 2Δm(X)S(X), which completes the proof.

In the following, we can obtain that Δm(X) can be used to characterize uniformly non-square space.

Proposition 4.7. Let X be a Banach space, then X is uniformly non-square if and only if Δm(X)>12.

Proof. Since X is not uniformly non-square if and only if ρ(1)=12 (see [4]). From the proof of Proposition 4.2, we can get that X is not uniformly non-square if and only if Δm(X)=12. This completes the proof.

Next, we will see that uniform non-squareness and ΔM(X) has a relationship.

Proposition 4.8. Let X be Banach space. If ΔM(X)<1, then X is uniformly non-square.

Proof. According to the proof of Proposition 4.2 and ΔM(X)<1, we can obtain δX(1)>0, and the proposition follows.

The converse of Proposition 4.8 is not true. Now, we provide a counterexample as follows:

Example 4.9. Let X be the space 2 endowed with the norm

x=(x1,x2)=(x1 ,x2 )1 ,x1 x2 0,(x1 ,x2 ) ,x1 x2 0.

Then ΔM(X)=1 and X is uniformly non-square.

Proof. From [26], Mizuguchi have proved that CNJ(X)=3+54<2, then X is uniformly non-square (see [31]).

However, fix x=(1,0), y=(0,1), then x-y=(1,-1). It is evident that x=y=xy=1. Now, from Proposition 4.2, we can obtain

1=12,12=x+y2ΔM(X)1,

which implies ΔM(X)=1.

### 4.2. The difference between the heights and midlines in equilateral triangles

In this section, in order to study the difference between the heights and midlines of equilateral triangles, we will introduce a new geometric constant.

Definition 4.10. For a given Banach space X, let

DΔ(X)=supx+yinfλ[0,1]λx+(1λ)y:x=y=xy=1.

Since λx+(1λ)y12 for any λ[0,1] and x=y=xy=1 by Theorem 1 in [27], then the definition of this constant is meaningful. The geometric meaning of DΔ(X) is the supremum of the ratio of twice the midlines to the heights, in equilateral triangles.

It is easy for us to obtain the equivalent form of DΔ(X):

DΔ(X)=supx+yλx+(1λ)y:x=y=xy=1,0λ1.

First, we give its bounds.

Proposition 4.11. Let X be a Banach space. Then 2DΔ(X)4.

Proof. Let λ=12, then for any x=y=xy=1, clearly

x+yλx+(1λ)y=2,

which implies the left inequality.

On the other hand, let x,ySX such that xy=1, we have x+yx+y=2. Then, according to Theorem 2 in [27], we can obtain λx+(1λ)y12, for any λ[0,1]. Thus, we can get

x+yλx+(1λ)y4.

This completes the proof.

The following result shows that the lower bound of DΔ(X) in the above proposition is sharp.

Proposition 4.12. Let X be a Hilbert space, then DΔ(X)=2.

Proof. Let x=y=xy=1. Assume that X is a Hilbert space, then

1=xy2=x22x,y+y2=22x,y.

Thus, we can obtain x,y=12. Then, we have

x+y2=x2+2x,y+y2=3.

And, by Proposition 3, we can get Δh(X)=32. Hence, we can obtain

DΔ(X)=supx+yλx+(1λ)y:x,ySX,xy=1,0λ1=3Δh(X)=2,

which completes the proof.

Remark 4.13. From Proposition 4.12, we can also obtain the heights coincide with the midlines for equilateral triangles in Hilbert spaces, which is the same as the statement of Corollary 4.5.

1. C. Alsina, P. Guijarro and M. Tomas, Some remarkable lines of triangles in real normed spaces and characterizations of inner product, Aequationes Math., 54(1997), 234-241.
2. J. Banaś, On moduli of smoothness of Banach spaces, Bull. Polish Acad. Sci. Math., 34(1986), 287-293.
3. D. Bárcenas, V. Gurariy, L. Sánchez and A. Ullán, On moduli of convexity in Banach Spaces, Quaest. Math., 27(2004), 137-145.
4. M. Baronti and P. L. Papini, Convexity, smoothness and moduli, Nonlinear Anal., 70(2009), 2457-2465.
5. G. Birkhoff, Orthogonality in linear metric spaces, Duke Math. J., 2(1)(1935), 169-172.
6. M. Brodskii and D. Milman, On the center of a convex set, Proc. Amer. Math. Soc., 59(1948), 837-840.
7. J. A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc., 40(3)(1936), 396-414.
8. J. A. Clarkson, The von Neumann-Jordan constant for the Lebesgue space, Ann. of Math., 38(1937), 114-115.
9. M. M. Day, Uniform convexity in factor and conjugate spaces, Ann. of Math., 45(1944), 375-385.
10. S. Dhompongsa, A. Kaewkhao and S. Tasena, On a generalized James constant, J. Math. Anal. Appl., 285(2)(2003), 419-435.
11. J. Gao, The uniform degree of the unit ball of a Banach spaces(I), Nanjing Daxue Xuebao. Ziran Kexue Ban, 1(1982), (in Chinese, English summary), 14-82 pp.
12. J. Garcĺa-Falset, E. Llorens-Fuster and E.M. Mazcuñan-Navarro, Uniformly nonsqure Banach spaces have the fixed point property for nonexpansive mappings, J. Funct. Anal., 233(2)(2006), 494-514.
13. K. Goebal, Convexity of balls and fixed-point theorems for mappings with nonexpansive square, Compos. Math., 22(3)(1970), 269-274.
14. V. I. Gurariy, On moduli of convexity and flattening of Banach spaces, Soviet Math. Dokl., 161(5)(1965), 1003-1006.
15. V. I. Gurariy and Y. U. Sozonov, Normed spaces that do not have distortion of the unit sphere, Mat. Zametki, 7(1970), 307-310.
16. C. He and Y. A. Cui, Some properties concerning Milman's moduli, J. Math. Anal. Appl., 340(2007), 1260-1272.
17. R. C. James, Uniformly non-square Banach space, Ann. of Math., 80(1964), 542-550.
18. R. C. James, Orthogonality in normed linear spaces, Duke Math. J., 12(1945), 291-302.
19. D. H. Ji and S. L. Wu, Quantitative characterization of the difference between Birkhoff orthogonality and isosceles orthogonality, J. Math. Anal. Appl., 323(2006), 1-7.
20. M. Kato, L. Maligranda and Y. Takahashi. On James and Jordan-von Neumann constants and the normal structure coefficient of Banach spaces, Stud. Math., 144(3)(2000), 275-295.
21. N. Komuro, K. S. Saito and R. Tanaka, Math. Nachr., 289(8-9)(2016), 1005-1020.
22. J. Lindenstrauss and L. Tzafriri. Classical Banach Spaces II: Function Spaces. 1st ed. Berlin, Germany: Springer; 1979.
23. H. Martini, K. J. Swanepoel and G. Weiß, The geometry of Minkowski spaces- a survey, I, Expo. Math., 19(2)(2001), 97-142.
24. V. D. Milman and G. Schechtman. Asymptotic theory of finite-dimensional normed spaces. Berlin, Germany: Springer; 1986.
25. H. Mizuguchi, The constants to measure the differences between Birkhoff and isosceles orthogonalities, Filomat, 30(10)(2016), 2761-2770.
26. H. Mizuguchi, The von Neumann-Jordan and another constants in Radon plane, Monatsh. Math., 195(2021), 307-322.
27. B. Z. Ni, C. He and D. H. Ji, A New Characterization of Uniformly Nonsquare Banach Spaces, Journal of Harbin University of Science and Techonlogy, 16(1)(2009), 107-109.
28. G. Nordlander, The modulus of convexity in normed linear spaces, Ark. Mat., 4(1)(1960), 15-17.
29. P. L. Papini and S. L. Wu, Measurements of differences between orthogonality types, J. Math. Anal. Appl., 397(2013), 285-291.
30. G. Pisier. The Volume of Convex Bodies and Banach Space Geometry. Cambridge, UK: Cambridge University Press; 1980.
31. Y. Takahashi and M. Kato, Von Neumann-Jordan constant and uniformly non-square Banach spaces, Nihonkai Math. J., 9(1998), 155-169.