1. 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
be a real inner product space. For a positive real number
s we denote
the s-ball of
, i.e.,
The Einstein addition
on
is a binary operation on
given by the equation
where
is the gamma factor
in
, where · and
are the inner product and the norm of
respectively. By the definition of
,
for every pair
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
.” 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
just for simplicity. The Einstein scalar multiplication
on
is given by the equation
where
; and
. By Theorem 6.84 in [
1],
is a gyrovector space, which is called an Einstein gyrovector space.
Ungar [
1] (pp. 172–173) defined the gyromidpoint
of two elements in a gyrovector space. For
we have
Ungar also defined the gyrocentroid
of three elements
as
The gyromidpoint corresponds to the average of two velocities in the special theory of relativity. On the other hand the gyrocentroid
does not satisfy a certain desirable property one would expected for
means; by a simple calculation we have
for
and
. 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].
2. The Metric Space
We define the set
which coincides with the open interval
.
admits the addition
and the scalar multiplication
given by the following:
where
and
. Please note that the triplet
is a real one-dimensional space.
The gyrometric is defined by
where
and
. The gyrometric needs not be a metric. It satisfies the following [
1] (p. 158).
Proposition 1. - (1)
For every pair , The equality holds if and only if .
- (2)
for any .
- (3)
The gyrotriangle inequality: holds for any in .
We define the metric on induced by the gyrometric d. Put the map by . For any and , the map f satisfies the following.
- (F1)
- (F2)
Let the map
on
be given by
for
.
Lemma 1. The inequalityholds for every pair . Proof. Recall the equations (3.177) and (3.178) [
1] (pp. 88–89):
By
we have
Since
then we have
Next we calculate
.
Thus, we have the desired inequalities and conclude the proof. □
Proposition 2. is a complete metric space.
Proof. We first prove that
is a metric space. By (1) and (2) of Proposition 1, it is trivial that
and
for every
. By (3) of Proposition 1 and the monotonicity of
f, the inequality
for every
. By (F1) we have
As
is complete, we have by Lemma 1 and the definition of
that
is complete. □
We recall the gyroline and the gyrosegment [
1] (Definition 6.19). Let
be elements of
. The gyroline through
and
is defined by
A gyrosegment with endpoints
and
is denoted by
The point is called the gyro t-point on a gyroline or gyrosegment. We abbreviate by . Please note that for every pair .
Theorem 1. For any we have 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
and
We also have
where
. Hence we have
Let
. It is well defined by
. We calculate the numerator of (8);
We conclude a proof of Theorem 1. □
By Theorem 1 and the monotonicity of
, we have
By the triangle inequality, we have
Moreover, since the map
is continuous, we infer that
g is convex, i.e.,
Letting
we have
3. 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 . We say that A is gyroconvex if for any . Let X be a non-empty subset of .We call the gyroconvex hull of X. Please note that the gyroconvex hull of a non-empty set is gyroconvex.
Lemma 2. Let . Then the gyrosegment is gyroconvex. The gyroconvex hull coincides with .
Proof. Let
be an arbitrary point in
for
. There exists
such that
for
. We may assume that
. where
. Then we have
,
. In fact, by the Equation (6.63) in [
1] (p. 167) we have
Since
, we have
. Thus,
for every pair
and
in
. Thus,
is gyroconvex. □
Let
be a non-empty subset of
. We define a sequence
of a non-empty subset of
by induction. Suppose that
is defined. Put
Proposition 3. Let be a non-empty subset of . Then Proof. We prove that is gyroconvex. Let . Since for every , there exists a positive integer with . Then by the definition of we have . Thus, is a gyroconvex set. As , we have .
We prove . For any , . Hence . Similarly, assuming that for any we have . So for arbitrary nonnegative integer n, ; . □
4. The Gyrogeometric Mean
Let . We define . First we prove the following.
Proposition 4. Suppose that . If the inequality holds for a positive real number M, then the inequality holds for arbitrary points and .
Proof. Put
and
, where
. We have
Applying this equality, we have by (
11) and (
12) that
□
Lemma 3. Let be a non-empty set. Then Proof. First we prove
. By Proposition 3,
where
. For any positive integer
k let
be arbitrary points in
. Then there exist
such that
. Put
then by Proposition 4.1 we have
whence
Thus, for arbitrary
,
. It follows that for any
, we have
Therefore
The converse inequality is trivial, hence we have
.
Next we prove . For any pair , there exist sequences and in such that converge to respectively. Letting for , we have . Thus, . The converse inequality is trivial, hence holds. □
Lemma 4. Suppose that K is a gyroconvex subset of . Then the closure of K is gyroconvex.
Proof. For any
, there exist
such that
converge to
respectively. We show
converges to
for arbitrary
. By (
11) we have
By
, then
. Thus,
. Hence
is gyroconvex. □
Let n be a positive integer. Let be the set of all subsets of whose number of elements is exactly n. We define, by induction, the sequence of the maps which satisfy the following two conditions () and ();
- ()
for every ,
- ()
for every pair and in .
First, put
for
. As
by Lemma 2 we obtain that
; (
) holds. Let
. Then by (
10) we have
which is (
).
Suppose now that the map
which satisfies (
) and (
) is defined. We will define
which satisfies (
) and (
). Let
. For a positive integer
m we define
which satisfies that
by induction on
m. For every
, put
Please note that
is well defined since
and we have assumed that the map
is defined. By the condition (
) we have that
for every
. Put
. Then
and
since
for every
. Suppose that
such that
is defined. For every
, put
As in the same way as the above,
is well defined for
, and
satisfies that
. Hence, by induction, we have defined a sequence
such that
. Applying (
) for
and
, we infer that
Then by Lemma 3 we obtain
for every positive integer
m. By Cantor’s intersection theorem there exists a unique
with
As
for every
, we infer that
for every
. Put
. Then the map
is well defined, and
;
satisfies the condition (
).
Next we prove that the map
satisfies the condition (
);
where
. Let
m be a positive integer. We define
and
for every
as in the same way as before. As (
) holds for
, we have
By summing up the above inequalities with respect to
we have
for every positive integer
m. Thus, we have
Letting
, since
, we have
hence
So, the condition (
) holds for the map
. We conclude that the map
which satisfies the conditions (
) and (
) are defined by induction.
By applying the maps we define the gyrogeometric mean of n elements in .
Definition 2. Let . We call that the gyrogeometric mean of Δ.
Due to the definition, the gyrogeometric mean of is . The gyrocentroid is defined by applying the internal division points on the usual lines which makes the inconvenience such as for . The gyrogeometric mean is defined by applying the gyrolines and it resolve the inconvenience.
5. 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 is the set of all n-points subset of for a positive integer n.
Proposition 5. Let and . Put . Then the following holds: Proof. We prove the equality (
13) by induction on
n. For
,
. By Theorem 6.37 in [
1] (p. 175) we have
Assume that (
13) holds for
. Let
where
for
. By the assumption we have
for every
. Then for
we have
We prove that
as
. By a simple calculation we have
as
. Hence we have
as
. Thus,
as
for
. We conclude that
. □
For and in , . More generally, the gyrogeometric mean satisfies the following.
Proposition 6. Let . In the case of
, it is proved by the following calculation.
Proposition 6 is proved by induction on
n.
In
Section 6 with appropriate operation is a gyrocommutative gyrogroup, which is also called the Einstein gyrogroup. The gyrogeometric mean is defined for
similarly. If
or
such that
is small enough,
. So, in the case,
is hold. It is simply proved by induction.
6. Proof that Is a Gyrocommutative Gyrogroup
A magma is a non-empty set G with a binary operation ⊕. A magma is a gyrogroup if its binary operation ⊕ satisfies the following axioms (G1) through (G5):
- (G1)
There exists a left identity
in
G such that
for all
.
- (G2)
For each
there exists a left inverse
such that
- (G3)
For any
there exists a unique element
such that the binary operation obeys the left gyroassociative law
- (G4)
The map given by is an automorphism of the magma . It is called a gyroautomorphism. generated by is called a gyration.
- (G5)
The gyroautomorphism
generated by any
satisfies the left loop property:
The gyrogroup is called gyrocommutative if the following (G6) holds for every pair
- (G6)
We prove that the Einstein gyrogroup 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
holds for all
. First, we prove the left cancellation law which is given by the equation
for all
. Put
for any
and put
Put
. We have
We compute,
and
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
is given by the following:
We will calculate each coefficient of
of the equation above.
We prove that the coefficient of
is 1, i.e.,
is 1. The equation
holds for all
. Applying this equation, we have
Multiplying
from the right-hand side of the last equation, and applying the gamma factor, we have
Dividing the common denominator
and multiplying
to
, where
, we have
We compute
of the Equation (
16).
Applying
for underline items, we infer that
So, we have
hence we have
.
Next, we prove that coefficient of is .
We have
. Then we compute the coefficient of
applying the equation
.
By
,
Finally, we prove that the coefficient of
is
.
Using (16) and
, then we have
Hence
holds. By applying the left cancellation law for the Equation (
14), we obtain (G3).
We prove (G4). We prove that
is automorphism for every pair
. To prove (G4), we first show the gyration preserves, the inner product of
and the norm. So, we compute
for all
. By applying the Equation (
15), we have
and
respectively, where
The terms
and
are defined in the similar way an
and
respectively. Then we have
We show that terms other than
of the right-hand side of the Equation (
18) equal to zero.
Then we compute each terms of the sum
.
By
this equation is rewritten in the following.
Then we obtain
Calculating
in a way similar to the calculation of
, we have
Hence, comparing the Equations (
19) with (
22) we have
We conclude that
.
To prove that
is a homomorphism for all
, we show
for all
. Applying (
15) we have
Put
By a simple calculation we infer that
We have
Then the right-hand side of the Equation (
24) is rewritten as the following equation.
Since
preserves the inner product and the norm of
, we have
so that
and
Hence
. We conclude that
is a homomorphism.
We observe that
is bijective for every pair of
. To prove this, we compute
for every
. We denote
where
. Then applying (
15), we have
We compute
and
So, we have
We show that the coefficients of
and
vanish. We compute the coefficient of
.
We also have
Applying the gamma identity, we have the following.
So, the coefficient of
vanishes.
We compute the coefficient of
.
We also have
So, the coefficient of
vanishes. Thus,
holds for every
. Changing
and
,
also holds for every
. We conclude that
is bijective. Thus,
is an automorphism; a proof of (G4) is complete.
To prove (G5) we first observe
for every pair
. Let
. Applying (
15) we have that
where
We prove that
for every
. We have
Then
is computed as in the following.
can be computed as in the following.
Hence
.
Next, we compute the coefficient of
.
can be computed as in the following.
The Equation (
25) is rewritten by
. Then