# Barycentre sur le bord de SL(3, R)/SO(3, R)

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28.09.2000

viii, 65 p

Thèse de doctorat: Université de Fribourg, 2000

English German In 1995, Besson, Courtois and Gallot [BCG] repeated the proof of a theorem that stated that any positive measure µ on the boundary M(∞) of a locally symmetric space M with rank 1 admits a unique center of mass as the solution of the equation: where Bθ is the Busemann function for θ ∈ M(∞). In the higher rank case it is obvious that the center of mass does not exist anymore. As an example we know that the center of mass on an Euclidean manifold, called a flat, does not exist. After the precise definition of the center of mass in chapter 3 the Cauchy boundary C is defined in chapter 4. The Cauchy boundary is a part of the usual boundary of SL(3,R)/SO(3,R) on which we can try to prove the existence of a center of mass: The existence and uniqueness of the center of mass for a discrete measure, that means for points, on the Cauchy boundary of SL(3,R)/SO(3,R) is given in chapter 5. We must add a slight restriction: If we consider a discrete measure on a rank 1 symmetric space we must have at least three points. In the higher rank case we generalize somewhat this condition. Points that satisfy this condition are called well-spread points. In chapter 6 there is an algorithm to compute that center of mass. The code is given for the cases where M = H2 and M = SL(3,R)/SO(3,R), as well as a list of examples. Some possible relation to statistics is given in chapter 7. More precisely we relate the center of mass of a family of points in R resp. R2 to an estimator in the family of Cauchy densities.
Faculty
Faculté des sciences
Department
Institut de Mathématiques
Language
• French
Classification
Mathematics
License
License undefined
Identifiers
Persistent URL
https://folia.unifr.ch/unifr/documents/299874
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