# Properties for the Frechet Mean in Billera-Holmes-Vogtmann Treespace

## TL;DR

This paper establishes necessary and sufficient conditions for edges to be in the Frechet mean within BHV treespace, aiding in its computation and advancing understanding of its geometric properties.

## Contribution

It provides the first precise edge inclusion criteria for the Frechet mean in treespace, enabling improved algorithms for its approximation.

## Key findings

- Edge inclusion conditions are characterized by inequalities on edge weights.
- These conditions serve as a pre-processing step for locating the Frechet mean.
- Results generalize to orthant spaces.

## Abstract

The Billera-Holmes-Vogtmann (BHV) space of weighted trees can be embedded in Euclidean space, but the extrinsic Euclidean mean often lies outside of treespace. Sturm showed that the intrinsic Frechet mean exists and is unique in treespace. This Frechet mean can be approximated with an iterative algorithm, but bounds on the convergence of the algorithm are not known, and there is no other known polynomial algorithm for computing the Frechet mean nor even the edges present in the mean. We give the first necessary and sufficient conditions for an edge to be in the Frechet mean. The conditions are in the form of inequalities on the weights of the edges. These conditions provide a pre-processing step for finding the treespace orthant containing the Frechet mean. This work generalizes to orthant spaces.