Some Results on Riemannean Multiple Barycenters

A characteristic of a special case of Riemannean barycenters on the unit circle is presented. The non-uniqueness of such barycenters leads to an interesting study of the so-called multiple barycenters. In this work, we deal with a smooth one-dimensional manifold S1 only. Some theoretical and computational analysis is listed.


Introduction
The Riemanean geometry is the study of curved surfaces.One might have a cylinder, or a sphere.For instance, one can use a cardboard paper towel roll to study a cylinder and a globe to study a sphere.In such spaces, the shortest curve between any pair of mass points on such curved surface is called a minimal geodesic or geodesic.One can find a geodesic between two mass points by stretching a rubber band between them.The first thing that will be noticed is that sometimes there is more than one minimal geodesic between two mass points.For instance, there are many minimal geodesics between the north and south poles of a sphere.This leads to derive more generalized formulas, which are true in the Euclidean space.Therefore, one of our main concern in this work is to motivate researcher for finding exact formulas for computing Barycenters on non-Euclidean spaces, namely on manifolds.In [1], the authors suggest an estimation of the so-called Riemannean Barycenters.For a use in the same context, we refer for instance to [2].Where, in [3] Riemannian means were presented as solution of variational problems.We think that for special case, for instance smooth manifolds, we could derive similar formulas to the Euclidean case, define and study interesting properties of Riemannean Barycenters.
In this work, we begin by setting out some basic theory of the geodesic of the unit circle as one dimensional manifold, we define the multiple Barycenters and compute it for some special cases.For more details, we refer to [4]- [6].
Let I be an interval of real numbers.Then, a curve γ is a continuous mapping given as we said that γ is a connection between x and y.We define ( ) the set of all connections between x and y on 1 S .Let x, y be two mass points of 1 S .The geodesic is the minimum distance between x and y on 1 S (or the shortest path), denoted by ( ) , g x y , which is given by ( ) We say that X is a geodesic space if for any pair , x y X ∈ there is a minimal geodesic from x to y and this is given as a minimal of ( ) , where ( ) where g is a geodesic on 1 S .For more details, we refer to [7]- [9].Example 1.Consider the following examples on the unit circle: 2) The midpoint of 3) Observe that even if . The fact that ( ) − is not a geodesic between x and y.
4) The midpoint on 1 S is not unique: Consider

Barycenters on 1 S
The arithmetic mean of a finite set of mass points 1 , , n x x  the Euclidean space d R . is the unique point x that minimizes the sum of the squared distances to the given mass point, that is ( ) , The point realizing the minimum of the function ( ) E x is called also Barycenter of the set of mass points ( ) x ≤ ≤ .In this section we would like to extend this definition to a smooth Riemannean manifold 1 S , and for more information, we refer to [1] [3].Definition 1.The Barycenter of the set of mass points 1 , , n x x  subset 1 S with weighs 1 , , 0 n w w ≥  (respectively) is the mass point minimizing the following functional (Energy function): where g(•,•) is the geodesic in 1 S and μ is the distribution in the continuous case.w , such that ( ) Proof.We prove that the geodesic Barycenter of two mass points is a mass point contained on the shortest path.We have In fact, the point c x satisfies the minimal and the following equality ( ) ( ) .
, conv , ; x y x x ∈ ( ) , g x y are unique, then the geodesic Barycenter of 1 2 , x x is unique.Proof.Suppose that the Barycenter is not unique, and denote by c x and c y solving the energy functional.From Lemma 1, we know that the geodesic Barycenter is contained on the shortest path between 1

Geodesic Multiple Barycenters on 1 S
The existence of Barycenters on 1 S interpreted as one dimensional Riemannean space does assure it uniqueness.In this section, we give a general condition of the existence of a geodesic multiple Barycenter on 1  S .Definition 3. (Geodesic Multiple Barycenter).We said that a set of mass points have a multiple Barycenter if the solution of the energy function is not unique.
1) The Barycenter of the north and south poles is not unique i.e. 0 and iπ e .2) We examine a simple example on the unit circle.Compute the Barycenter of four mass points min : min ,0 ,e ,e ,e .
x S x S F x g x g x g x g x 2) x S ∈ with ( ) 3) x S ∈ with ( ) We conclude that iπ 4 i3π 4 i5π 4 i7π 4 e , e , e and e 4 are Barycenters of the set X. Definition 4. (Symmetry propriety SP).Let µ be a mass distribution on the unit circle.The map  [ ] , for which θ satisfies the map ( θ θ = = .If the distribution is defined over the set of equidistant masses with is A a θ θ -rotation on the unit circle, with a supplemen- tary propriety that 1, , , 1 shows some example of symmetry mass points on 1 . S (From left to right) On the first figure, we have two mass points antipodal, the first on (1, 0) Cartesian coordinate and the second in (0, 1).The second, third and fourth show other types of semmetries.
Theorem 1 (Multiple Barycenter).Let S be a set of n mass points.If there exists a symmetry map A θ , then X admits a multiple Barycenter.

θ =
A θ is the symmetry map such that ( ) ( ) , and , .Proof.The proof is left to the reader.Figure 2 contains three, four, five, six, eight and nine mass points satisfying the symmetry property and non necessary with same weights.The geodesic between two arbitrary and successive mass points is constant.I each of the figures our code generates all possible Barycentes so-called multiple Barycenters.
Figure 3 shows the uniqueness of the Barycenter on 1 S in the case of four masses without symmetry property.

Concluding Remarks
As consequence of the non-uniqueness geodesic between two arbitrary mass points, we have presented an example on the notion of multiple Barycenters.Moreover, we have derived some properties similarly to the Euclidean space.We believe that more interesting cases are to construct significant formulas for arbitrary manifolds.
Riemannean manifold.The curve γ is said to be simple if it is injective.If I is a closed bounded interval [a, b], we also allow the possibility ( ) ( ) makes it possible to talk about closed simple curve).If ( ) ( ) the extremities of I), then ( ) x γ is called a double (or multiple) point of the curve.A curve  is said to be closed or a loop if I = [a, b] and if ( ) ( ) curve is thus a continuous mapping of the circle.Let x, y are two points of 1 S subset of n R .If γ is a curve such that ( ) ∈ are two mass points with weights 1 w and 2

2 w
respectively.If for all

2 ..
contradicts the uniqueness of the geodesic.Definition Let an oriented unit circle with origin We denote by 0 w the number of all geodesics, their corresponding curves contain the origin symmetry of µ if: σ is an isometry and ( ) .

Lemma 3 .
Under the same assumptions of Theorem 1.Let k be the number of the symmetry maps A θ in  1 which have the propriety A X X θ = , i.e. the distribution of the mass points is A θ -invariant.The number of the multiple Barycenters is k + 1.

Figure 3 .
Figure 3. Case of non-symmetry on S 1 (uniqueness of the Barycenter.(small perturbation of the north and south poles respectively.