 
Summary: MEDIANS AND MEANS IN FINSLER GEOMETRY
MARC ARNAUDON AND FRANK NIELSEN
Abstract. We investigate existence and uniqueness of pmeans 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 those Finsler center points.
Contents
1. Introduction 1
2. Preliminaries 3
3. Forward pmeans 7
4. Forward median 10
5. An algorithm for computing pmeans 12
References 14
1. Introduction
The geometric barycenter of a set of points is the point which minimizes the
sum of the squared distances to these points. It is the most traditional estimator is
statistics that is however sensitive to outliers [17]. Thus it is natural to replace the
average distance squaring (power 2) by taking the power of p for some p [1, 2).
