Convexity in Tree Spaces
Bo Lin, Bernd Sturmfels, Xiaoxian Tang, Ruriko Yoshida · SIAM Journal on Discrete Mathematics · 2017
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 ${CAT}(0)$ 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 exhibit properties that are desirable for geometric statistics, such as geodesics of small depth.