A Remark on the Homogeneity of Isosceles Orthogonality

Throughout this paper, X = (X, ‖ ⋅ ‖) always denotes a real normed linear space with origin o, unit ball B X , and unit sphere S X . The dimension of such a space is always assumed to be at least 2. Isosceles orthogonality, an extension of the orthogonality in inner product spaces to normed linear spaces, was introduced by James in [1]. Two vectors (or points) x and y in X are said to be isosceles orthogonal (denoted by x⊥ I y) if and only if


Introduction
Throughout this paper,  = (, ‖ ⋅ ‖) always denotes a real normed linear space with origin , unit ball   , and unit sphere   .The dimension of such a space is always assumed to be at least 2.
Isosceles orthogonality, an extension of the orthogonality in inner product spaces to normed linear spaces, was introduced by James in [1].Two vectors (or points)  and  in  are said to be isosceles orthogonal (denoted by ⊥  ) if and only if      +      =      −      .
In the same paper James proved that the implication characterizes inner product spaces.In other words, isosceles orthogonality is homogeneous if and only if the underlying space is an inner product space.Although implication (2) does not hold in a normed linear space which is not an inner product space, it might hold "locally." For example, let  be the normed linear space on R 2 with the maximum norm and  = (1, 1).Then In fact, in this example one can easily verify that ⊥   if and only if  is a multiple of (−1, 1).The existence of vectors satisfying implication (3) motivated the authors of [2] to introduce the following definition.
Definition 1.A unit vector  in  such that implication (3) holds is called a homogeneous direction of isosceles orthogonality.
We denote by   the set of all homogeneous directions of isosceles orthogonality.Note that if  is an inner product space then   =   and that   might be an empty set for some normed linear space  (cf.Example 2.1 in [1]).
In the book [3], Amir mentioned the following implication similar to (2) which also characterizes inner product spaces (cf.[3, (4.12)] and [4, I 1 )]): there exists a number  ∈ (0, 1) such that ,  ∈ , A related characterization can be found in [3, (10.13)] (or [5]).By modifying (4) a little bit, we can introduce the notion of almost homogeneous direction of isosceles orthogonality.Note that, in the following definition, we do not need the number  to be chosen uniformly.
Definition 2. Let  be a unit vector in a normed linear space .If for each  ∈ , which is isosceles orthogonal to , there exists a number  := (, ) ∈ (0, 1) such that ⊥  , then  is called an almost homogeneous direction of isosceles orthogonality.
We denote by    the set of all almost homogeneous directions of isosceles orthogonality.It is clear that We will show that, surprisingly, the converse of ( 5) is also true.

Main Result
We will use also the notion of Birkhoff orthogonality.A vector  is said to be Birkhoff orthogonal (see [6,7]) to another vector  if the inequality ‖ + ‖ ≥ ‖‖ holds for each real number .For relations between isosceles orthogonality and Birkhoff orthogonality and related results, we refer to the survey [8].
Lemma 3 (cf.Section 3 in [7] or Theorems 4.8 and 4.9 in [8]). Let Then one can verify that In the following we show that  = , which will complete the proof.
By the construction of  there exists a real number  0 such that  =  +  0 .Since  ∈    , there exists a sequence {  } ∞ =1 of positive numbers, which is strictly decreasing and converges to 0, such that ⊥     for each integer  ≥ 1. ) . ( Then it follows from (10) that  0 = 0. Therefore  = , as claimed.
Our main result is the following theorem.Moreover, it is clear that  is the unique point  in  + ⟨−, ⟩ such that (, ) + (, ) = 0. Lemma 5 and the existence of a point in +⟨−, ⟩ that is isosceles orthogonal to  show that ⊥  , and the proof is complete.

Corollary 7. Let 𝑋 be a Banach space whose dimension is at least two. If the relative interior of 𝐻 󸀠
in   is not empty, then  is a Hilbert space.