Local Approximate Symmetry of Birkhoff–James Orthogonality in Normed Linear Spaces

TL;DR: In this article, a local approximate symmetry of the Birkhoff-James orthogonality in normed linear spaces has been investigated, and it has been shown that the orthogoneality is approximately symmetric in the sense of Dragomir and Chmielinski in all finite-dimensional Banach spaces.

Abstract: Two different notions of approximate Birkhoff–James orthogonality in normed linear spaces have been introduced by Dragomir and Chmielinski. In the present paper we consider a global and a local approximate symmetry of the Birkhoff–James orthogonality related to each of the two definitions. We prove that the considered orthogonality is approximately symmetric in the sense of Dragomir in all finite-dimensional Banach spaces. For the other case, we prove that for finite-dimensional polyhedral Banach spaces, the approximate symmetry of the orthogonality is equivalent to some newly introduced geometric property. Our investigations complement and extend the scope of some recent results on a global approximate symmetry of the Birkhoff–James orthogonality.

Summary

1. Introduction

• The Birkhoff–James orthogonality is the most natural and well studied notion of orthogonality in normed linear spaces.
• It should be noted that the authors of [5] considered this notion in the global sense, the meaning of which will be clear once the authors present the relevant definition in this section.
• Now, with respect to the Dragomir’s definition, the authors define the following analogous versions of approximate symmetry considered in Definitions 1.1 and 1.2.
• The authors show that for any finite-dimensional polyhedral Banach space with property (P1), local property (P) also holds for each element.

4. C-approximate Symmetry for Two-Dimensional Polyhedral Banach Spaces

• (P1) We first prove that in any finite-dimensional polyhedral Banach space property (P1) always implies the local property (P) for each x ∈ SX .the authors.the authors.
• This contradicts that the local property (P) fails for x and thus the result follows.
• Applying Theorem 3.14, the authors now prove that in any two-dimensional regular polyhedral Banach space with at least 6 vertices, the Birkhoff–James orthogonality is C -approximately symmetric.
• Let the ordinate meet the boundary of the polygon at ±(0, β).

Declarations

• The research of Dr. Divya Khurana and Dr. Debmalya Sain is sponsored by Dr. D. S. Kothari Postdoctoral Fellowship under the mentorship of Professor Gadadhar Misra.
• The authors declare that they have no conflict of interest.
• Availability of data and material Not applicable.

Figures

Figure 5. Nonpolyhedral, not approximately symmetric 2- ∞ space with R(X) = 1.

Figure 6. 3-dimensional space with property (P) but not

Figure 8. Two-dimensional regular polyhedral space with 8

Figure 7. Two-dimensional regular polyhedral space with 6

Figure 1. Approximate left-symmetry - not approximate

Figure 2. Approximate right-symmetry - not approximate

Figure 9. Two-dimensional not regular polyhedral space.

Figure 4. Property (P) and R(X) > 1.

Figure 3. Property (P) and R(X) > 1.

Full text

Results Math (2021) 76:136
c
 2021 The Author(s), under exclusive licence to
Springer Nature Switzerland AG
1422-6383/21/030001-26
published online June 17, 2021
https://doi.org/10.1007/s00025-021-01437-y
Results in Mathematics
Local Approximate Symmetry of
Birkhoff–James Orthogonality in Normed
Linear Spaces
Jacek Chmieli´nski, Divya Khurana, and Debmalya Sain
Abstract. Two different notions of approximate Birkhoff–James orthog-
onality in normed linear spaces have been introduced by Dragomir and
Chmieli´nski. In the present paper we consider a global and a local ap-
proximate symmetry of the Birkhoff–James orthogonality related to each
of the two definitions. We prove that the considered orthogonality is ap-
proximately symmetric in the sense of Dragomir in all finite-dimensional
Banach spaces. For the other case, we prove that for finite-dimensional
polyhedral Banach spaces, the approximate symmetry of the orthogo-
nality is equivalent to some newly introduced geometric property. Our
investigations complement and extend the scope of some recent results
on a global approximate symmetry of the Birkhoff–James orthogonality.
Mathematics Subject Classification. Primary 46B20; Secondary 51F20,
52B15, 47L05.
Keywords. Birkhoff–James orthogonality, approximate Birkhoff–James
orthogonality, C-approximate symmetry, D-approximate symmetry.
1. Introduction
The Birkhoff–James orthogonality is the most natural and well studied notion
of orthogonality in normed linear spaces. In general, the Birkhoff–James or-
thogonality is not symmetric. Chmieli´nski and W´ojcik [5] introduced a notion
of approximate symmetry of the Birkhoff–James orthogonality in normed lin-
ear spaces. It should be noted that the authors of [5] considered this notion
in the global sense, the meaning of which will be clear once we present the

136 Page 2 of 26 J. Chmieli´nski et al. Results Math
relevant definition in this section. In this article, our motivation is to consider
the corresponding local version of the aforesaid concept. We also study the
local version of another standard notion of an approximate Birkhoff–James
orthogonality considered in [6]. The advantage of considering the local version
is illustrated by obtaining some useful conclusions in the global case, separately
for finite-dimensional polyhedral Banach spaces and smooth Banach spaces.
Let us first establish the notations and the terminologies to be used in the
present article. Throughout the text, we use the symbols X, Y to denote real
normed linear spaces. Given any two elements x, y ∈ X,let
xy =conv{x, y} =
{(1 − t)x + ty : t ∈ [0, 1]} denote the closed line segment joining x and y.By
B
X
= {x ∈ X : x≤1} and S
X
= {x ∈ X : x =1} we denote the unit ball
and the unit sphere of X, respectively, and B(x, δ) denotes the open unit ball
in X centered at x and with the radius δ>0. The collection of all extreme
points of B
X
will be denoted as Ext B
X
.
Let X
∗
denote the dual space of X. Given 0 = x ∈ X, f ∈ S
X
∗
is said to
be a supporting functional at x if f (x)=x.LetJ(x)={f ∈ S
X
∗
: f(x)=
x},0= x ∈ X, denote the collection of all supporting functionals at x. Note
that for each 0 = x ∈ X, the Hahn-Banach theorem ensures the existence of
at least one supporting functional at x.
An element x ∈ S
X
is said to be a smooth point if J (x)={f} for some
f ∈ S
X
∗
.LetsmS
X
denote the collection of all smooth points of S
X
.In
particular if sm S
X
= S
X
, then X is said to be a smooth space.LetX be a
Banach space with a norm . For every τ>0, the modulus of smoothness is
defined by
ρ(τ)=sup

x + τy+ x −τy−2
2
: x, y ∈ S
X

.
(X, ) is said to be a uniformly smooth space if lim
τ→0
ρ(τ)
τ
=0.
Let X be a Banach space with a norm . For every ε ∈ (0, 2], the
modulus of convexity is defined by
δ(ε)=inf

1 −
x + y
2
: x, y ∈ B
X
, x − y≥ε

.
(X, ) is said to be uniformly convex if δ(ε) > 0 for all ε ∈ (0, 2].
It is well known that a Banach space (X, ) is uniformly smooth if and
only if its dual space (X
∗
, 
∗
) is uniformly convex (see [10] for more details).
For x, y ∈ X, we say that x is Birkhoff–James orthogonal to y [2,7],
written as x ⊥
B
y,ifx + λy≥x for all λ ∈ R.In[7, Theorem 2.1],
James proved that if 0 = x ∈ X, y ∈ X, then x ⊥
B
y if and only if there exists
f ∈ J(x) such that f (y) = 0. We will use the notations x
⊥
= {y ∈ X : x ⊥
B
y}
and
⊥
x = {y ∈ X : y ⊥
B
x}. Sain [12] characterized the Birkhoff–James
orthogonality of linear operators between finite-dimensional Banach spaces by
introducing the notions of the positive part of x, denoted by x
+
, and the
negative part of x, denoted by x
−
, for an element x ∈ X. For any element

Vol. 76 (2021) Local Approximate Symmetry Page 3 of 26 136
y ∈ X, we say that y ∈ x
+
(y ∈ x
−
)ifx + λy≥x for all λ ≥ 0(λ ≤ 0).
It is easy to see that x
⊥
= x
+
∩ x
−
.
Dragomir [6] defined an approximate Birkhoff–James orthogonality as
follows. Let ε ∈ [0, 1) and let x, y ∈ X; then x is said to be approximately
Birkhoff–James orthogonal to y if x+λy≥(1−ε)x for all λ ∈ R. Later on,
Chmieli´nski [3] slightly modified the definition given by Dragomir as follows.
Let ε ∈ [0, 1) and let x, y ∈ X. Then x is said to be approximately Birkhoff–
James orthogonal to y, written as x ⊥
ε
D
y, if and only if x+λy≥
√
1 − ε
2
x
for all λ ∈ R. Due to this modification, in case of a Hilbert space, the present
notion of the approximate orthogonality coincides exactly with the usual no-
tion of the ε-orthogonality: |x, y| ≤ εxy.In[9, Lemma 3.2], Mal et al.
proved that
x ⊥
ε
D
y ⇔∃f ∈ S
X
∗
: |f (x)|≥

1 − ε
2
x and f(y)=0. (1.1)
Chmieli´nski [3] defined a variation of approximate Birkhoff–James or-
thogonality. Given x, y ∈ X and ε ∈ [0, 1), x is said to be approximately
orthogonal to y, written as x ⊥
ε
B
y,ifx + λy
2
≥x
2
− 2εxλy for
all λ ∈ R. Later, in [4, Theorems 2.2 and 2.3], Chmieli´nski et al. gave two
characterizations of this approximate orthogonality:
x ⊥
ε
B
y ⇔∃z ∈ span{x, y} : x ⊥
B
z, and z − y≤εy; (1.2)
x ⊥
ε
B
y ⇔∃f ∈ J(x): |f (y)|≤εy. (1.3)
Given x, y ∈ X and ε ∈ [0, 1), we will write x ⊥
ε
∗
D
y (x ⊥
ε
∗
B
y)ifx ⊥
ε
D
y
(x ⊥
ε
B
y) but x ⊥
ε
1
D
y (x ⊥
ε
1
B
y) for any 0 ≤ ε
1
<ε.
In general, the orthogonality relation between two elements x, y ∈ X
need not be symmetric. In other words, for any two elements x, y ∈ X, x ⊥
B
y
does not necessarily imply y ⊥
B
x. James [8]provedthatifdimX ≥ 3and
the Birkhoff–James orthogonality is symmetric, then the norm is induced by
an inner product. For more details on the recent study of these notions of
approximate Birkhoff–James orthogonality see [14,15].
In [5], Chmieli´nski and W´ojcik defined the following notion of approxi-
mate symmetry of the Birkhoff–James orthogonality in a normed linear space.
Definition 1.1 Let X be a normed linear space. Then the Birkhoff–James or-
thogonality is approximately symmetric if there exists ε ∈ [0, 1) such that
whenever x, y ∈ X and x ⊥
B
y, it follows that y ⊥
ε
B
x.
The above definition is global in the sense that ε is independent of x and
y.
In this paper we will work with both of the above mentioned notions
of approximate Birkhoff–James orthogonality. To avoid any confusion we will
call the above notion of approximate symmetry an approximate symmetry
of the Birkhoff–James orthogonality in the sense of Chmieli´nski or shortly:
C-approximate symmetry of the Birkhoff–James orthogonality.

136 Page 4 of 26 J. Chmieli´nski et al. Results Math
In [5], the authors gave an example of a Banach space where the Birkhoff–
James orthogonality is not C-approximately symmetric. In the present article
we will study this example in more detail. The following definition allows us
to study local versions of the C-approximate symmetry of the Birkhoff–James
orthogonality.
Definition 1.2 Let X be a normed linear space and let x ∈ X. We say that x
is C-approximately left-symmetric (C-approximately right-symmetric) if there
exists ε
x
∈ [0, 1) such that whenever y ∈ X and x ⊥
B
y (y ⊥
B
x), it follows
that y ⊥
ε
x
B
x (x ⊥
ε
x
B
y).
For A⊆X we say that the Birkhoff–James orthogonality is
C-approximately symmetric on A if there exists ε ∈ [0, 1) such that when-
ever x, y ∈Aand x ⊥
B
y, it follows that y ⊥
ε
B
x.
Let A⊆X and let x ∈ S
X
. We say that x is C-approximately left-
symmetric (C-approximately right-symmetric) on A if there exists ε
x
∈ [0, 1)
such that whenever y ∈Aand x ⊥
B
y (y ⊥
B
x), it follows that y ⊥
ε
x
B
x
(x ⊥
ε
x
B
y).
Now, with respect to the Dragomir’s definition, we define the following
analogous versions of approximate symmetry considered in Definitions 1.1 and
1.2.
Definition 1.3 Let X be a normed linear space. We say that the Birkhoff–
James orthogonality is approximately symmetric in the sense of Dragomir,
shortly: the Birkhoff–James orthogonality is D-approximately symmetric,if
there exists ε ∈ [0, 1) such that whenever x, y ∈ X and x ⊥
B
y, it follows that
y ⊥
ε
D
x.Forx ∈ X, we define x to be D-approximately left-symmetric (D-
approximately right-symmetric), if there exists ε
x
∈ [0, 1) such that whenever
y ∈ X and x ⊥
B
y (y ⊥
B
x), it follows that y ⊥
ε
x
D
x (x ⊥
ε
x
D
y).
Observe that we can restrict ourselves to norm-one elements by virtue of
the homogeneity of all the notions of orthogonality and approximate orthogo-
nality introduced here.
To study the C-approximate left-symmetry and the C-approximate right-
symmetry of elements of a normed linear space X, we define the following
property. We say that the local property (P) holds for x ∈ S
X
if
x
⊥
∩ A (x)=∅,
where A (x) is the collection of all those elements y ∈ S
X
for which given any
f ∈ J(y), either f or −f is in J(x).
We say that the property (P) holds for a normed linear space X if the
local property (P) holds for each x ∈ S
X
,thatis,
for all x ∈ S
X
: the local property (P) holds. (P)
If A⊆S
X
and x ∈ S
X
, then we say that the local property (P) holds for
x on A if x
⊥
∩ A (x) ∩A= ∅.

Vol. 76 (2021) Local Approximate Symmetry Page 5 of 26 136
It follows trivially that the local property (P) holds for each x ∈ sm S
X
.
We will prove that the local property (P) for an x ∈ S
X
is equivalent to the
C-approximate left-symmetry of x in the local sense, that is, the local property
(P) holds for x ∈ S
X
if and only if for y ∈ x
⊥
∩ S
X
there exists ε
x,y
∈ [0, 1)
such that y ⊥
ε
x,y
B
x.
To study polyhedral Banach spaces, we recall the following definitions
from [13] which are relevant to our work:
Definition 1.4 Let X be an n-dimensional Banach space. A polyhedron P is
a non-empty compact subset of X which is an intersection of finitely many
closed half-spaces of X, that means P = ∩
r
i=1
M
i
, where M
i
are closed half-
spaces in X and r ∈ N.Thedimension of the polyhedron P is defined to be
the dimension of the subspace generated by the differences x − y of vectors
x, y ∈ P .
An n-dimensional Banach space X is said to be a polyhedral Banach space
if B
X
contains only finitely many extreme points, or, equivalently, if S
X
is a
polyhedron.
Definition 1.5 Let X be an n-dimensional Banach space. A polyhedron Q ⊆ X
is said to be a face of the polyhedron P ⊆ X if either Q = P or if we can
write Q = P ∩ δM, where M is a closed half-space in X containing P and
δM denotes the boundary of M . If the dimension of Q is i, then Q is called
an i-face of P .(n − 1)-faces are called facets of P and 1-faces of P are called
edges of P .
Definition 1.6 Let X be a finite-dimensional polyhedral Banach space and let
F be a facet of the unit ball B
X
. A functional f ∈ S
X
∗
is said to be a supporting
functional corresponding to the facet F of the unit ball B
X
if the following
two conditions are satisfied:
(a) f attains its norm at some point v of F .
(b) F =(v + ker f) ∩ S
X
.
It is easy to see that there is a unique hyperspace H such that an affine
hyperplane parallel to H contains the facet F of the unit ball B
X
.Moreover,
there exists a unique norm-one functional f , such that f attains its norm on
F and ker f = H. In particular, f is a supporting functional to B
X
at every
point of F .
Two elements x, y ∈ Ext B
X
of an n-dimensional polyhedral Banach
space X are said to be adjacent if tx +(1− t)y = 1 for all t ∈ [0, 1].
Given normed linear spaces X, Y ,byB(X, Y )(K(X, Y )) we denote the
space of all bounded (compact) linear operators from X to Y . A bounded linear
operator T ∈B(X, Y ) is said to attain its norm at x ∈ S
X
if Tx = T .Let
M
T
= {x ∈ S
X
: Tx = T } be the collection of all norm attaining elements
of T .IfX is a reflexive Banach space and T ∈K(X, Y ), then M
T
= ∅ (see [1]
for details).

Frequently asked questions

Q1. What contributions have the authors mentioned in the paper "Local approximate symmetry of birkhoff–james orthogonality in normed linear spaces" ?

In the present paper the authors consider a global and a local approximate symmetry of the Birkhoff–James orthogonality related to each of the two definitions. The authors prove that the considered orthogonality is approximately symmetric in the sense of Dragomir in all finite-dimensional Banach spaces. For the other case, the authors prove that for finite-dimensional polyhedral Banach spaces, the approximate symmetry of the orthogonality is equivalent to some newly introduced geometric property. 

Q2. What is the only supporting linear functional for y?

The only supporting linear functional for y is the supporting functional g ∈ SX∗ corresponding to the edge v2v3 such that g(x) = 1 for all x ∈ v2v3. 

Q3. What is the definition of a n-dimensional Banach space?

An n-dimensional Banach space X is said to be a polyhedral Banach space if BX contains only finitely many extreme points, or, equivalently, if SX is a polyhedron. 

Q4. what does the property hold for a normed linear space?

The authors say that the property (P) holds for a normed linear space X if the local property (P) holds for each x ∈ SX , that is,for all x ∈ SX : the local property (P) holds. 

Q5. What is the local property of a two-dimensional polyhedral Banach space?

1. Consider a twodimensional polyhedral Banach space X = R2, whose unit sphere is determined by the extreme points v1 = (2, 2), v2 = (1, 3), v3 = (0, 3.5), v4 = (−1, 3), v5 = (−2, 2), v6 = −v1, v7 = −v2, v8 = −v3, v9 = −v4, v10 = −v5. 

Q6. What is the main motivation behind considering the local property (P) for x S?

Recall that the local property (P) holds for x ∈ SX ifx⊥ ∩ A (x) = ∅, where A (x) is the collection of all those elements y ∈ SX for which given any f ∈ J(y), either f or −f is in J(x). 

Q7. What is the definition of the Birkhoff–James orthogonality?

In particular, the authors prove that in all finite-dimensional Banach spaces, the Birkhoff–James orthogonality is always D-approximately symmetric. 

Q8. What is the modulus of convexity for x?

For every ε ∈ (0, 2], the modulus of convexity is defined byδ(ε) = inf {1 − ‖x + y‖ 2: x, y ∈ BX , ‖x − y‖ ≥ ε } .(X, ‖ ‖) is said to be uniformly convex if δ(ε) > 0 for all ε ∈ (0, 2]. 

Q9. What is the simplest way to denote the dual space of X?

Let X∗ denote the dual space of X. Given 0 = x ∈ X, f ∈ SX∗ is said to be a supporting functional at x if f(x) = ‖x‖. Let J(x) = {f ∈ SX∗ : f(x) = ‖x‖}, 0 = x ∈ X, denote the collection of all supporting functionals at x. 

Q10. What is the definition of the dimension of the polyhedron P?

The dimension of the polyhedron P is defined to be the dimension of the subspace generated by the differences x − y of vectors x, y ∈ P . 

Q11. What is the advantage of considering the local version of the Birkhoff–James ortho?

The advantage of considering the local version is illustrated by obtaining some useful conclusions in the global case, separately for finite-dimensional polyhedral Banach spaces and smooth Banach spaces. 

Q12. What is the simplest way to define a smooth space?

It is well known that a Banach space (X, ‖ ‖) is uniformly smooth if and only if its dual space (X∗, ‖ ‖∗) is uniformly convex (see [10] for more details). 

Q13. What is the characterization of the Birkhoff–James orthogonality?

Then any y ∈ SX is C-approximately right-symmetric if and only if the property (P) holds for X.Now, the authors present a complete characterization of the C-approximate symmetry of the Birkhoff–James orthogonality in finite-dimensional polyhedral Banach spaces.