Circle-Uniqueness of Pythagorean Orthogonality in Normed Linear Spaces

We denote byX = (X, ‖ ⋅ ‖) a real normed linear space whose dimension is at least 2.The origin, unit ball, and unit sphere of X are denoted by o, B X , and S X , respectively. WhenX is twodimensional, it is called aMinkowski plane. Its unit sphere S X is then called the unit circle of X, and each homothetic copy of S X is a circle. For two distinct points (or vectors) x and y in X, we denote by ⟨x, y⟩ the line passing through x and y, by [x, y⟩ the ray starting from x and passing through y, and by [x, y] the (nondegenerate) segment connecting x and y. Moreover,X is said to be strictly convex if S X does not contain a nondegenerate segment. Pythagorean orthogonality, which was introduced by James in [1], is one of the most natural extensions of orthogonality in inner product spaces to normed linear spaces (for other orthogonality types in normed linear spaces, we refer to [2–4] and the references therein). Let x and y be two vectors in a real normed linear space. If


Introduction
We denote by  = (, ‖ ⋅ ‖) a real normed linear space whose dimension is at least 2. The origin, unit ball, and unit sphere of  are denoted by ,   , and   , respectively.When  is twodimensional, it is called a Minkowski plane.Its unit sphere   is then called the unit circle of , and each homothetic copy of   is a circle.For two distinct points (or vectors)  and  in , we denote by ⟨, ⟩ the line passing through  and , by [, ⟩ the ray starting from  and passing through , and by [, ] the (nondegenerate) segment connecting  and .Moreover,  is said to be strictly convex if   does not contain a nondegenerate segment.
Pythagorean orthogonality, which was introduced by James in [1], is one of the most natural extensions of orthogonality in inner product spaces to normed linear spaces (for other orthogonality types in normed linear spaces, we refer to [2][3][4] and the references therein).Let  and  be two vectors in a real normed linear space.If then  and  are said to be Pythagorean orthogonal to each other (denoted by ⊥  ).James showed that the following facts are equivalent: (1) ,  ∈ ,  ∈ R, ⊥   ⇒ ⊥  ; (2)  is an inner product space.
In other words, Pythagorean orthogonality is not homogeneous in general normed linear spaces.Among other things, James proved the line-existence of Pythagorean orthogonality: for each pair of vectors  and  in , there exists a number  such that ⊥   + .That is, James proved that in each line parallel to the line ⟨−, ⟩ there exists a vector that is Pythagorean orthogonal to .However, James did not obtain any essential result on the uniqueness of this orthogonality type.Kapoor and Prasad [5] fixed this gap by proving that Pythagorean orthogonality is line-unique in each normed linear space , where a binary relation ⊥ on  is said to be line-unique if and only if for each  ̸ =  and  ∈  there exists a unique real number  such that  ⊥  + .It appears that the uniqueness of Pythagorean orthogonality has nothing to do with the shape of the unit ball.By introducing the circle-uniqueness (see Definition 1 next) of Pythagorean orthogonality, we show that this is not true.Our main result shows that Pythagorean orthogonality is circle-unique if and only if  is strictly convex, which updates the knowledge about uniqueness of Pythagorean orthogonality.
For each  ∈ , we denote by () the set of points that are Pythagorean orthogonal to ; that is, For two linearly independent vectors  and  we denote by  , the two-dimensional subspace of  spanned by  and  and by  , the closed halfplane of  , bounded by the line ⟨−, ⟩ and containing .
Definition 1. Pythagorean orthogonality on  is said to be circle-unique if, for each pair of linearly independent vectors  and  and each nonnegative real number , there exists a unique vector  in   ∩  , ∩ ().

Results and Proofs
The following lemma concerning the intersection of two circles in a Minkowski plane is one of our main tools.
Lemma 2 (Theorem 2.4 in [6]).Let  1 :=  1   +  1 and  2 :=  2   +  2 be two circles in a Minkowski plane , where  1 and  2 are two distinct points, and let  and  be the points of intersection of ⟨ 1 ,  2 ⟩ and  1 .Then the set  1 ∩  2 has one of the following forms: First we show that Pythagorean orthogonality has the circle-existence property.More precisely, we show the following proposition.Proposition 4. For each pair of linearly independent vectors  and  and each number  ≥ 0, the set is a nonempty segment that may degenerate to a singleton.
Proof.We only consider the nontrivial case  > 0. Clearly, Since is not empty.It is also clear that   ∩ ⟨−, ⟩ = 0. Thus, by Lemma 2,   is the union of two closed, disjoint segments contained in  , , one or both of which may degenerate to a singleton, lying in opposite halfplanes with respect to the line ⟨−, ⟩.This completes the proof.
Next we state a simple result on common supporting lines of two circles.Lemma 5. Let  be a Minkowski plane,  ̸ =  a vector in , and ,  > 0 two numbers such that 0 <  −  < ‖‖ . ( Then there are two common supporting lines of   and   +  passing through the point  = (/( − )).
Proof.By the hypothesis of the lemma,  is exterior to   .Thus two supporting lines  1 and  2 of   can be drawn through .In the following we show that these two lines are two common supporting lines of   and   + .
Clearly, there exists a point  ∈   such that  1 supports   at .Put  0 = ( − )/.Then that is, the distance from  to  1 is ‖‖ = .Thus  1 is a common supporting line of   and   + .In a similar way we can show that  2 is also a common supporting line of these two discs.
which implies that ⟨, ⟩ is one of the two common supporting lines of   and   + .Lemma 5 shows that ⟨, ⟩ intersects ⟨−, ⟩ at .Then there exist two unit vectors  and V such that Thus there exists a number  ∈ (0, 1) such that  =  + (1 − )V.Since [ + , V + ] is the unique maximal segment contained in (  + ) ∩  , and parallel to [, ], the lines ⟨, ⟩ = ⟨, V⟩ and ⟨ + , V + ⟩ coincide.From In the rest of the proof we show that the intersection of the segments [, V] and [+, V+] is a nontrivial segment, which forces the set  to be a nondegenerate segment.It suffices to show that  +  is a relatively interior point of the segment [, V].
On the one hand, we have