Medians and means in Finsler geometry

TL;DR

This paper studies the existence, uniqueness, and computation of p-means and medians on Finsler manifolds, introducing gradient flow methods and algorithms for finding these central points.

Contribution

It establishes the existence and uniqueness of p-means and medians in Finsler geometry and proposes a convergent algorithm based on gradient flows for their computation.

Findings

Proved p-means are limit points of gradient flows.

Discretized gradient flows converge to p-means under certain conditions.

Provided an algorithm for computing Finsler medians and means.

Abstract

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