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Medians and means in Finsler geometry

Part of: Local differential geometry

Published online by Cambridge University Press:  01 February 2012

Marc Arnaudon  and
Frank Nielsen
Marc Arnaudon
Affiliation:
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)
Frank Nielsen
Affiliation:
Laboratoire d’Informatique (LIX), École Polytechnique, 91128 Palaiseau Cedex, France Sony Computer Science Laboratories, Inc, Tokyo, Japan (email: frank.nielsen@acm.org)

Abstract

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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.

MSC classification

Secondary: 53B40: Finsler spaces and generalizations (areal metrics)
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 15 , May 2012 , pp. 23 - 37
Copyright
Copyright © London Mathematical Society 2012

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