# Convexity in Tree Spaces

## Bernd Sturmfels, UC Berkey

## April 26, 2016

We discuss the geometry of metrics and convexity structures on the
space of phylogenetic trees, 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 and can have arbitrarily high dimension. Tropical
convexity and the tropical metric are better behaved, as they exhibit
properties that are desirable for geometric statistics.

This is joint work with Bo Lin, Xioaxian Tang and Ruriko Yoshida.