LMS Journal of Computation and Mathematics

Table of Contents - Volume 15 - 2012  

Research Article

Medians and means in Finsler geometry

Marc Arnaudona1 and Frank Nielsena2a3

a1 Laboratoire de Mathématiques et Applications, CNRS: UMR 6086, Université de Poitiers, Téléport 2 – BP 30179, F-86962 Futuroscope, Chasseneuil Cedex, France (email: marc.arnaudon@math.univ-poitiers.fr)

a2 Laboratoire d’Informatique (LIX), École Polytechnique, 91128 Palaiseau Cedex, France

a3 Sony Computer Science Laboratories, Inc, Tokyo, Japan (email: frank.nielsen@acm.org)

Abstract

We investigate existence and uniqueness of p-means ep and the median e1 of a probability measure μ on a Finsler manifold, in relation with the convexity of the support of μ. We prove that ep is the limit point of a continuous time gradient flow. Under some additional condition which is always satisfied for p≥2, a discretization of this path converges to ep. This provides an algorithm for determining the Finsler center points.

(Received November 26 2010)

(Revised September 11 2011)

(Online publication February 2012)

2000 Mathematics Subject Classification

  • 53B40 (primary)