Convexity in Tree Spaces

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 $\mathrm{CAT}(0)$-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.

21 pages, 5 figures; Theorem 13 is now proved in all dimensions

References in corpus (4)

• Duality and separation theorems in idempotent semimodules
• Principal components analysis in the space of phylogenetic trees
• The space of ultrametric phylogenetic trees
• An algorithm for constructing principal geodesics in phylogenetic treespace

Cited by in corpus (7)

• Tropical Fermat-Weber points
• Tropical medians by transportation
• Tropical Geometric Variation of Phylogenetic Tree Shapes
• Tropical convexity in location problems
• An Invitation to Tropical Alexandrov Curvature
• Tropical Geometric Tools for Machine Learning: the TML package
• Convex geometry over ordered hyperfields