1. Introduction
Recently, many investigations have been devoted to geometric constants of a Banach space X, among which, the Jordan–von Neumann constant and the James constant have been widely studied, and their results have greatly motivated us to make specific descriptions of a variety of geometric properties of X.
Recall a classic constant
proposed by M. Baronti, E. Casini, P.L. Papini in 2000 [
1] that manifested antipodal points
y and
on the unit sphere
, where
and 2 could be considered as the lengths of the sides of the triangle with vertices
lying on
. It is noteworthy that the constant
describes the supremum of the arithmetic mean of the lengths of the changeable sides of the triangle, i.e.,
Alonso and Llorens-Fuster [
2] have discussed the relation between
and
as
Strongly motivated by the asymmetric constant
proposed by Gao in 2000 [
3], Chen et al. has further investigated the constant
proposed by Gao by discussing several properties of it that have not yet been discovered. A constant
closely relevant to
, along with a variety of geometric properties and several relations among it and several basic geometric constants, has also been shown via a few inequalities. For more details about the aforementioned results, we recommend reference [
4].
In this paper, we intend to conduct further research on the related constants combined with the Baronti constant. Firstly, we mainly focus on a new Baronti type constant
in
Section 3, which combines the unit sphere of the Banach spaces with the inscribed equilateral triangles and innovatively takes the quantity
into consideration. Then, we want to bring out a few connections among its value and some geometrical properties of the space. Additionally, the relations among
and some other constants such as
will be revealed by the clarification of several inequalities.
can measure how large the sum of two distances from a point of the inscribed equilateral triangle to the midpoint of the opposite side can be. In other terms, their values hinges on the lengths of two midlines of a triangle with one vertex at the center of the unit ball and the other on the unit sphere. Moreover, some inequalities enable us to continue introducing some other new constants
,
and
. It is interesting to study the non-symmetrical properties of these constants and hopefully obtain some significant results.
2. Preliminaries
Throughout the paper, let X be a non-trivial Banach space, that is, , and use and to represent the unit sphere and closed unit ball of X, respectively.
Definition 1. A Banach space X is called uniformly non-square if there exists such that for any , we have either or .
The geometric constant below was proposed by Yang et al. [
5]:
Definition 2. Let X be a Banach space. The function is defined by Clearly the following assertion was also shown in the paper [5]: J. Lindenstrauss [
6] has studied several properties of the following modulus of smoothness of
X.
Definition 3. Let X be a Banach space. The modulus of smoothness is defined by X is called uniformly smooth if .
In addition, we recall the definitions of some main geometric constants as follows.
Definition 4. - (i)
The James constant (Gao [7], Gao and Lau [8]): - (ii)
The Jordan–Von Neumann constant (Clarkson [9]): - (iii)
The hexagon constant (Gao [3]): - (iv)
The Baronti constant (Baronti et al. [1]):
Now we enumerate several results of the aforementioned constants. Gao has conducted a study of
with the following properties in 2000 [
3]:
- (i)
For a Banach space, ;
- (ii)
If X is a Hilbert space, then ;
- (iii)
Let or . Let be defined as usual. Then, for ; for .;
- (iv)
If , then X is uniformly non-square.
Investigations on the relations among
,
and
can be found in [
1,
10,
11,
12], along with their common geometric properties.
- (i)
, , ;
- (ii)
If X is a Hilbert space, then , a fortiori ;
- (iii)
X is uniformly non-square if and only if one of the following is true: (a) , (b) , (c) .
3. Some Inequalities Related to New Constant
The heuristic idea in this paper comes from a study of the height lines of triangle, which were defined as the height vector
, and then provided to give the new characterizations of inner product spaces [
13]. Highly inspired by the study of height lines of triangle and its discussion of properties in real normed space, we will explore the properties of midlines of triangle via discussion of the new constant
along with its properties. Firstly, we define the constant
of a Banach space
X as follows:
The geometric background of
is shown in
Figure 1: consider the unit sphere in the Euclidean plane with
. Assume that
,
. Then,
. Assume that
C is the midpoint of
, and
D is the midpoint of
. Then,
and
. Apparently, the geometric concept of
is the supremum of sum of two midlines.
Proposition 1. If X is an inner product space, then .
Proof. For any
,
and
Therefore, . □
Proposition 2. Let X be a finite-dimensional Banach space. If , then X is not rotund.
Proof. Since
X is a finite-dimensional Banach space with
, there exist
, such that
so
thus
hence
Combining with , , we have By and ; therefore, we obtain that X is not rotund. □
Proposition 3. Let X be Banach space. Then
Proof. We can let
, and deduce
, which implies that
. So
that is,
□
Proposition 4. Let X be a Banach space. Then
Proof. We can let
, and deduce that
which implies that
. Therefore,
that is,
□
Proof. For
, we have
which gives
.
Since
we obtain the left-hand side of the inequality. □
Theorem 1. Let X be a Banach space. If , then X is uniformly non-square.
Proof. Applying Proposition 5, we have
. Utilizing the result in ([
3], p. 243, Theorem 2.10), then
X is uniformly non-square. □
Corollary 1. Let X be a Banach space. If , then X has the fixed point property.
Proof. Utilizing Theorem 1 and [
14], every uniformly non-square Banach space has the fixed point property, so we get the result as required. □
Proposition 6. Let X be a Banach space. Then .
Proof. For any
, we have
and then
.
and so
. □
Example 1. Let endowed with the norm for . Then, .
Assume that ; then, .
Then, . Since , so .
Example 2. Let X be endowed with the norm defined by Assume that
; then
. Then,
and then we have
4. Some Results Related to New Constants and
For any Banach space X, we define another constant , which may be different from in view of its value. We also want to bring out the relations among and some geometric constants, along with its geometric properties.
Proposition 7. If X is an inner product space, then .
Proof. For any
,
and
Therefore, . □
Proof. For
, we have
which gives
.
Since , we obtain the left-hand side of the inequality. □
Theorem 2. Let X be a Banach space. If , then X is uniformly non-square.
Proof. Applying Proposition 8, we have . Utilizing Theorem 1, then X is uniformly non-square. □
Next, we consider the relation between and through inequality.
Proof. For any
, we have
Therefore, we complete the proof. □
Additionally, for the sake of further survey on the upper bounds of
, we utilize a parameter
, which is defined as follows:
Proof. For any
, we have
Thus, we complete the proof. □
Theorem 5. For any Banach space X, if , then .
Proof. For any
, there exist
such that
Since
, and
, it can be deduced that
Now that
can be arbitrarily small, then
Combined with the fact that
we have
. □
Now we shall consider the generalised form of , which is defined as follows.
Proposition 9. If X is an inner space, then .
Proof. For any
,
and
Thus, . □
Example 3. Let X be , . Then, .
Applying Clarkson’s type inequality, when
, we have
We also use the following Clarkson’s inequality
Then we can deduce that for any
,
as desired.
Example 4. Let X be endowed with the norm defined by Assume that
; then
. Then,
and we thus have
Therefore,
and then we have
Note that the values of the subduplicate form of the aforementioned constants can also be different. We therefore consider another new constant , which is denoted as follows:
Proposition 10. If X is an inner product space, then .
Proof. For any
,
and
Therefore . □
Proof. Since
is concave, then for any
, we have
which gives
.
Since , we obtain the left-hand side of the inequality. □
Theorem 6. Let X be a Banach space. If , then X is uniformly non-square.
Proof. Applying Proposition 11, we have
. Utilizing the result in ([
3], p. 243, Theorem 2.10), then
X is uniformly non-square. □
Example 5. Let X be endowed with the norm defined by Assume that
; then
. Then,
and then we have
5. Conclusions
In light of the asymmetric geometric constant proposed by Gao and the Baronti type constant , we intend to introduce a new geometric constant , which measures the lengths of two midlines of a triangle with one vertex at the center of the unit ball and the other on the unit sphere. In this paper, we manage to show several geometric properties, such as rotundity and the uniform nonsquare property, via characterizing its relationships with a variety of other basic geometric constants. Moreover, we provide a study of the quadratic form and subduplicate form of this constant, and , respectively, by means of exploration of their relations with some well-known geometric constants. However, there are still plenty of interesting problems that await discussion. How can , and be utilized to characterize more geometric properties? More results will be presented in our future research for the reader interested in going further in geometric constants of a Banach space.