Secondary fans of hypersimplices and finite metric spaces

The secondary fan of a polytope stratifies the regular subdivisions of the polytope. We compute this fan for the hypersimplices $\Delta(2,7)$ and $\Delta(3,6)$. We also find new families of rays of the secondary fan of $\Delta(k,n)$ which do not lie on the respective Dressians.
In the special case $k=2$ our results are closely related to metric spaces on $n$ points.

This is joint work with Michael Joswig and Lars Kastner.