# Convexity in Tree Spaces

arXiv: https://arxiv.org/abs/1510.08797 · HTML: https://mlsanity.com/paper/1510.08797/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Oct 2015.

## Built on by (most cited first, 1 of 1)

- [Tropical Geometric Tools for Machine Learning: the TML package](https://mlsanity.com/paper/2309.01082/) (2023)

## Abstract (arXiv)

We study the geometry of metrics and convexity structures on the space of phylogenetic trees, which is here realized as the tropical linear space of all  ultrametrics. The rm CAT(0)-metric of Billera-Holmes-Vogtman arises from the theory of orthant spaces. While its geodesics can be computed by the Owen-Provan algorithm, geodesic triangles are complicated. We show that the dimension of such a triangle can be arbitrarily high. Tropical convexity and the tropical metric behave better. They exhibit properties desirable for geometric statistics, such as geodesics of small depth.