^{1}

^{1}

Let

When we try to imagine or picture a reflection of a point

However, we all know that, in general, these “mirror symmetries” do not hold when we work inside a Banach space. So naturally, one should wonder about the types of spaces that do possess such symmetries. Our aim in this paper is to give necessary and sufficient conditions for this to be true and to investigate the consequences of these symmetries on the geometry of the unit ball of the Banach space; see Section

Throughout this section,

Let

We also need the following definition.

Let

It is clear that, in general, an element

Let

(i) An element

(ii) An element

(iii) Every element in

It follows directly from Definitions

To explain the use of the term “weak” symmetry in Definition

Let

(i) The Banach space

(ii) The subspace

(iii) For every

(i)

(ii)

It follows from the previous lemma that one can find a Banach space

A Banach space

The notions of weak-reflection and weak symmetry that were discussed above are quite different from the “usual” notions of reflection and symmetry. As mentioned above, when we think about a reflection point

Let

Setting

If

We recall the following notion of orthogonality, which was first introduced by Roberts [

Two elements

Two nonempty subsets

If

Note that this definition of orthogonality is symmetric in the sense that

The following observation gives an alternative way of thinking about the notion of symmetry introduced in Definition

Let

To address the question of uniqueness of reflection points (equivalently of orthogonal projections), we recall the following definition.

Let

First we take a look at the uniqueness of centers. We have the following.

If

Suppose

Now let

The following follows from the proof of the previous lemma.

Let

We note that there are various notions of centers in the literature and that they all play an important role in approximation theory and in geometric functional analysis in general.

We have seen above in Section

Let

(i) If

(ii) Every element

(iii) Every element

(i) Suppose that

(ii) Let

(iii) This follows directly from Part (ii) and Remark

The following follows immediately from Theorem

A Banach space

For every element

It is important to note that

Consider the Banach space

We also point out that the existence of a center of

Let

If

If

We now highlight the effect of symmetry on the geometry of the unit ball

If a Banach space

Since

One should note that the orthogonal symmetry of the unit ball given by Theorem

Consider again the example given in Example

We now consider the case where

Suppose that a Banach space

Let

The reader may be wondering about the situation when a Banach space

Let

Suppose

Since, by Remark

Let

With these definitions in mind, we have the following.

If

First we prove that

Note that Theorems

Let

We note that there are symmetries in

Then we have the following.

Let

First note that, since

To prove that

Note that

We conclude with some suggestions for further investigations:

With Theorem

Instead of a general Banach space, we may consider specific Banach spaces

As mentioned above in Introduction, there are many results in the literature that rely either directly or indirectly on the geometry of the unit ball of a Banach space. It may be interesting to revisit these and investigate the presence of orthogonal symmetries and their consequences.

The authors declare that they have no conflicts of interest.