A Gyrogeometric Mean in the Einstein Gyrogroup

: In this paper, we deﬁne a gyrogeometric mean on the Einstein gyrovector space. It satisﬁes several properties one would expect for means. For example, it is permutation-invariant and left-translation invariant. It is already known that the Einstein gyrogroup is a gyrocommutative gyrogroup. We give an alternative proof which depends only on an elementary calculation.


Introduction
Einstein addition is a binary operation that stems from his velocity composition law of relativistic admissible velocities. Einstein addition is neither commutative nor associative. Ungar initiated the study of gyrogroups and gyrovector spaces [1] associated with the Einstein addition in the theory of special relativity. A gyrocommutative gyrogroup is a gyrogroup which has weak associativity and commutativity. It is a generalization of a commutative group.
Let V be a real inner product space. For a positive real number s we denote V s the s-ball of V, i.e., The Einstein addition ⊕ E on V s is a binary operation on V s given by the equation where γ u is the gamma factor in V s , where · and · are the inner product and the norm of V respectively. By the definition of ⊕ E , u ⊕ E v ∈ V s for every pair u, v ∈ V s by Theorem 3.46 and the identity (3.189) in [1].
In [1] (p. 88) Ungar described that "one can show by computer algebra that Einstein addition in the ball is a gyrocommutative gyrogroup operation, giving rise to the Einstein ball gyrogroup (V 1 , ⊕ E )." On the other hand, Suksumran and Wiboonton [2] gave a proof applying the theory of Clifford algebras, without using computer algebras. We give an elementary and direct proof in Section 6, which is lengthy but just by a simple calculation without applying any substantial theory of mathematics.
In the following up to Section 5 we assume s = 1 just for simplicity. The Einstein scalar multiplication ⊗ E on (V 1 , ⊕ E ) is given by the equation where r ∈ R, a ∈ V 1 \ {0}; and r ⊗ E 0 = 0. By Theorem 6.84 in [1], (V 1 , ⊕ E , ⊗ E ) is a gyrovector space, which is called an Einstein gyrovector space. Ungar [1] (pp. 172-173) defined the gyromidpoint P m ab of two elements in a gyrovector space. For a, b ∈ V 1 we have Ungar also defined the gyrocentroid C m abc of three elements a, b, c ∈ V 1 as The gyromidpoint corresponds to the average of two velocities in the special theory of relativity. On the other hand the gyrocentroid C m abc does not satisfy a certain desirable property one would expected for means; by a simple calculation we have C m abc = γ c γ c +2 c = 1 3 ⊗ E c for a = b = 0 and c = 0. In this paper, we propose an alternative definition of the mean of three or more elements, the gyrogeometric mean, and show that it has several properties one would expect for means. The gyrogeometric mean corresponding to the average of the velocities in the special relativity. It is symmetric in the sense that permutation-invariant by the definition of the gyrogeometric mean. It is translation invariant (Proposition 5). The main idea of the definitions come from the geometric mean for positive definite matrices by Bhatia and Holbrook [3] and Ando, Li and Mathias [4].

The Metric Space
We define the set V 1 = {± v : v ∈ V 1 } ⊂ R which coincides with the open interval (−1, 1). V 1 admits the addition ⊕ and the scalar multiplication ⊗ given by the following: where a, b ∈ V 1 and r ∈ R. Please note that the triplet ( V 1 , ⊕ , ⊗ ) is a real one-dimensional space.
The gyrometric is defined by where a, b ∈ V 1 and a E b = a ⊕ E (−b). The gyrometric needs not be a metric. It satisfies the following [1] (p. 158).
The gyrotriangle inequality: holds for any a, b, c in V 1 .
We define the metric δ on V 1 induced by the gyrometric d. Put the map f : V 1 → R by f (x) = tanh −1 (x). For any a, b ∈ V 1 and r ∈ R, the map f satisfies the following.
Let the map δ on V 1 be given by for a, b ∈ V 1 .
we have Since a · b ≤ a b then we have Thus, we have the desired inequalities and conclude the proof.
Proof. We first prove that (V 1 , δ) is a metric space. By (1) and (2) of Proposition 1, it is trivial that δ(a, b) = 0 ⇔ a = b and δ(a, b) = δ(b, a) for every a, b ∈ V 1 . By (3) of Proposition 1 and the monotonicity of f , the inequality f (d(a, b) As V is complete, we have by Lemma 1 and the definition of δ(·, ·) that (V 1 , δ) is complete.
We recall the gyroline and the gyrosegment [1] (Definition 6.19). Let a, b be elements of V 1 . The gyroline through a and b is defined by A gyrosegment with endpoints a and b is denoted by is called the gyro t-point on a gyroline or gyrosegment. We abbreviate a# 1 2 b by a#b. Please note that a#b = b#a for every pair a, b ∈ V 1 .
Proof. To begin with the proof of the inequality (2), we show an equation related to the gyrometric and gamma factor. Recall the equations (3.197) and (6.266) [1] (pp. 93, 209): Hence holds. By a simple calculation, we have Hence we have γ (a#b) E (a#c) = γ a#b γ a#c (1 − (a#b) · (a#c)) and We also have We calculate the numerator of (8); We conclude a proof of Theorem 1.

The Gyroconvex Set and the Gyroconvex Hull in a Gyrovector Space
We define a gyroconvex set and a gyroconvex hull.

Definition 1.
Let A be a subset of V 1 . We say that A is gyroconvex if S(a, b) ⊂ A for any a, b ∈ A. Let X be a non-empty subset of V 1 .
We call conv(X) the gyroconvex hull of X.
Let C 0 be a non-empty subset of V 1 . We define a sequence {C n } of a non-empty subset of V 1 by induction. Suppose that C n−1 is defined. Put Proof. We prove that ∪ ∞ k=0 C n is gyroconvex. Let a, b ∈ ∪ ∞ k=0 C n . Since C k ⊂ C k+1 for every k ∈ N ∪ {0}, there exists a positive integer n 0 with a, b ∈ C n 0 . Then by the definition of C n 0 +1 we have S(a, b) ⊂ C n 0 +1 ⊂ ∪ ∞ k=0 C n . Thus, ∪ ∞ n=0 C n is a gyroconvex set. As C 0 ⊂ ∪ ∞ k=0 C n , we have conv(C 0 ) ⊂ ∪ ∞ n=0 C n .

Lemma 4.
Suppose that K is a gyroconvex subset of V. Then the closure K of K is gyroconvex.

Proof.
For any x, y ∈ K, there exist {x n }, {y n } ⊂ K such that x n , y n converge to x, y respectively. We show x n # t y n converges to x# t y for arbitrary 0 ≤ t ≤ 1. By (11) we have By n → ∞, then δ(x n # t y n , x#y) → 0. Thus, x# t y ∈ K. Hence K is gyroconvex.
Let n be a positive integer. Let X n be the set of all subsets of V 1 whose number of elements is exactly n. We define, by induction, the sequence {G n } ∞ n=2 of the maps G n : X n → V 1 which satisfy the following two conditions (p n ) and (q n ); . . , a n } and ∆ = {a 1 , . . . , a n } in X n .
Please note that a 1 i is well defined since {a 0 1 , . . . , a 0 i−1 , a 0 i+1 , . . . , a 0 k+1 } ∈ X k and we have assumed that the map G k is defined. By the condition (p k ) we have that As in the same way as the above, a l+1 Then by Lemma 3 we obtain for every positive integer m. By Cantor's intersection theorem there exists a unique M ∞ ∈ V 1 with Next we prove that the map G k+1 satisfies the condition (q k+1 ); . Let m be a positive integer. We define a m i and a m i for every 1 ≤ i ≤ k + 1 as in the same way as before. As (q k ) holds for G k , we have By summing up the above inequalities with respect to ).
for every positive integer m. Thus, we have j , a j )).
So, the condition (q k+1 ) holds for the map G k+1 . We conclude that the map G n : X n → V 1 which satisfies the conditions (p n ) and (q n ) are defined by induction. By applying the maps G n we define the gyrogeometric mean of n elements in V 1 .
Due to the definition, the gyrogeometric mean of {a, b} ⊂ V 1 is a#b. The gyrocentroid C m abc is defined by applying the internal division points on the usual lines which makes the inconvenience such as C m abc = γ c γ c +2 c = 1 3 ⊗ E c for a = b = 0. The gyrogeometric mean is defined by applying the gyrolines and it resolve the inconvenience.

Properties of the Gyrogeometric Mean
The gyrogeometric mean satisfies certain desirable properties one would expect for means in general. For example, the permutation invariance and the left-translation invariance would be expected properties. It is trivial that the gyrogeometric mean is permutation-invariant. We prove that the gyrogeometric mean is left-translation invariant.
Recall that X n is the set of all n-points subset of V 1 for a positive integer n.
We prove that x ⊕ E D m k+1 → {x ⊕ E G k+1 (D k+1 )} as m → ∞. By a simple calculation we have More generally, the gyrogeometric mean satisfies the following.
In the case of n = 2, it is proved by the following calculation.
Proposition 6 is proved by induction on n.
In Section 6 V s = {v ∈ V : v < s} with appropriate operation is a gyrocommutative gyrogroup, which is also called the Einstein gyrogroup. The gyrogeometric mean is defined for V s similarly. If s → ∞ or v ∈ V s such that v is small enough, γ v → 1. So, in the case, G n ({a 1 , a 2 , · · · , a n }) → a 1 + a 2 + · · · + a n n is hold. It is simply proved by induction.

Proof that (V s , ⊕ E ) Is a Gyrocommutative Gyrogroup
A magma (G, ⊕) is a non-empty set G with a binary operation ⊕. A magma (G, ⊕) is a gyrogroup if its binary operation ⊕ satisfies the following axioms (G1) through (G5): There exists a left identity 0 in G such that For each a ∈ G there exists a left inverse a ∈ G such that a ⊕ a = 0.
(G3) For any a, b, c ∈ G there exists a unique element gyr[a, b]c ∈ G such that the binary operation obeys the left gyroassociative law We prove that the Einstein gyrogroup (V s , ⊕ E ) is in fact a gyrocommutative gyrogroup only by simple calculations. Proof of (G1) and (G2) are simple and omitted.
We prove (G3). We prove that u ⊕ E (v ⊕ E w) = (u ⊕ E v) ⊕ E gyr[u, v]w holds for all u, v, w ∈ V s . First, we prove the left cancellation law which is given by the equation for all a, b ∈ V s . Put D uv = 1 + u·v s 2 for any u, v ∈ V s and put We compute, Hence we have Next, we prove the following equation It is known in [5] ((2.84), (2.85)) that the Equation (14) can be rewritten as by applying computer algebra, where We prove (15) without applying computer algebra. Put Put also Then gyr [u, v]w is given by the following: We will calculate each coefficient of u, v, w of the equation above.
We prove that the coefficient of w is 1, i.e., D zy D ux D vw γ u γ v γ z is 1. The equation a·b s 2 = D ab − 1 holds for all a, b ∈ V s . Applying this equation, we have Multiplying D ux D vw from the right-hand side of the last equation, and applying the gamma factor, we have Dividing the common denominator D uv γ u We compute {·} of the Equation (16). Applying for underline items, we infer that So, we have Next, we prove that coefficient of u is A D .
We have 1 + γ z = 1 + D uv γ u γ v = D. Then we compute the coefficient of u applying the equation Finally, we prove that the coefficient of v is B D .
Using (16) and D zy D ux D vw γ z = 1 γ u γ v , then we have Hence gyr[u, v]w = w + Au+Bv D holds. By applying the left cancellation law for the Equation (14), we obtain (G3).
We prove (G4). We prove that gyr[u, v] is automorphism for every pair u, v ∈ V. To prove (G4), we first show the gyration preserves, the inner product of V and the norm. So, we compute for all a, b, u, v ∈ V s . By applying the Equation (15), we have · a), The terms A b and B b are defined in the similar way an A a and B b respectively. Then we have By u 2 s 2 = γ u 2 −1 γ u 2 this equation is rewritten in the following.
Then we obtain Calculating B a B b v 2 in a way similar to the calculation of A a A b u 2 , we have Hence, comparing the Equations (19) with (22) we have · a)).
We conclude that gyr[u, v]a · gyr[u, v]b = a · b.
To prove that gyr[u, v] is a homomorphism for all u, v ∈ V s , we show for all x, y ∈ V s . Applying (15) we have By a simple calculation we infer that We have Then the right-hand side of the Equation (24) where a, b, c ∈ V s . Then applying (15), we have We compute (u) and D can be computed as in the following.
D uv (1+γ u ) = A D . Next, we compute the coefficient of v. B can be computed as in the following.

The Equation (25) is rewritten by
Hence 1 D (A 1 D uv γ u + B ) = B D . We conclude that gyr[u ⊕ E v, v]w = gyr[u, v]w for all u, v, w ∈ V s , so (G5) holds.
We conclude that (V s , ⊕ E ) is a gyrogroup. Finally, we prove that it is gyrocommutative. We prove (G6). We prove that a ⊕ b = gyr[a, b](b ⊕ a) for all a, b ∈ V s . Gyroautomorphic inverse property defined by Ungar in [1] (Definition 3.1, p. 51) is given by the equation where a, b are arbitrary elements in V s . According to Theorem 3.2 in [1] (p. 51), (V s , ⊕ E ) is gyrocommutative if and only if it has the gyroautomorphic inverse property. So, we observe the gyroautomorphic inverse property: Hence (V s , ⊕ E ) is gyrocommutative.
Author Contributions: Conceptualization of gyrogeometric mean, T.H. The proof that the Einstein gyrogroup is a gyrogroup is by T.H. Except these all authors contributed equally to this article. All authors have read and agreed to the published version of the manuscript.
Funding: This research was funded by the Japan Society for the Promotion of Science: 19K03536.

Conflicts of Interest:
The authors declare no conflict of interest.