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A Remark on the Homogeneity of Isosceles Orthogonality
Inspired by the definition of homogeneous direction of isosceles orthogonality, we introduce the notion of almost homogeneous direction of isosceles orthogonality and show that, surprisingly, these two notions coincide. Several known characterizations of inner products are improved.
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 .
Isosceles orthogonality, an extension of the orthogonality in inner product spaces to normed linear spaces, was introduced by James in . 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 with the maximum norm and . Then In fact, in this example one can easily verify that if and only if is a multiple of . The existence of vectors satisfying implication (3) motivated the authors of  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 ).
In the book , Amir mentioned the following implication similar to (2) which also characterizes inner product spaces (cf. [3, (4.12)] and [4, I1)]): A related characterization can be found in [3, (10.13)] (or ). 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 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.
2. 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 .
Therefore, a point satisfies if and only if Clearly, the midpoint of the segment is the unique point satisfying .
One can easily verify the following lemma.
Lemma 4. Let and be two linearly independent vectors in and a positive number. Then
Lemma 5. If , then, for each point that is isosceles orthogonal to , .
Proof. We only consider the nontrivial case when . Set
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 such that . Since , there exists a sequence of positive numbers, which is strictly decreasing and converges to , such that for each integer . Thus we have the equality or, equivalently, When is sufficiently close to , one can verify the following: By combining (13) with the triangle inequalities we obtain from which it follows that
Similarly, when is sufficiently close to , we have the equality
From (16) and the triangle inequalities it follows that From (17) it follows that By adding (15) and (18), we obtain Then it follows from (10) that . Therefore , as claimed.
Our main result is the following theorem.
Theorem 6. If , then .
Proof. Let be an arbitrary point that is isosceles orthogonal to and an arbitrary number. We only need to consider the case . By Lemma 5, Thus Moreover, it is clear that is the unique point in such that . Lemma 5 and the existence of a point in that is isosceles orthogonal to show that , and the proof is complete.
Corollary 7. Let be a Banach space whose dimension is at least two. If the relative interior of in is not empty, then is a Hilbert space.
Conflict of Interests
The authors declare that they have no conflict of interests to this work.
Chan He is supported by the Scientific Research Foundation of Graduate School of Harbin University of Science and Technology (Grant no. HLGYCX2011-008), a foundation from the Ministry of Education of Heilongjiang Province (Grant no. 1251H013) and the National Nature Science Foundation of China (Grant nos. 11371114 and 11171082).
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Copyright © 2014 Chan He and Dan Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.