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arXiv · 1510.08797

Convexity in Tree Spaces

Bo Lin·Bernd Sturmfels·Xiaoxian Tang·Ruriko Yoshida
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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 ${\rm CAT}(0)$-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.

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Bo Lin, Bernd Sturmfels, Xiaoxian Tang, Ruriko Yoshida. 2016-06-14. Convexity in Tree Spaces. https://doi.org/10.1137/16m1079841

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