Convexity in Tree Spaces
Metric Geometry
2018-02-19 v3 Computational Geometry
Combinatorics
Populations and Evolution
Abstract
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 -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.
Cite
@article{arxiv.1510.08797,
title = {Convexity in Tree Spaces},
author = {Bo Lin and Bernd Sturmfels and Xiaoxian Tang and Ruriko Yoshida},
journal= {arXiv preprint arXiv:1510.08797},
year = {2018}
}
Comments
21 pages, 5 figures; Theorem 13 is now proved in all dimensions