Subdivisions of Hypersimplices: with a View Toward Finite Metric Spaces

The secondary fan $\mathrm{\Sigma }(k,n)$ is a polyhedral fan which stratifies the regular subdivisions of the hypersimplices $\mathrm{\Delta }(k,n)$. We find new infinite families of rays of $\mathrm{\Sigma }(k,n)$, and we compute the fans $\mathrm{\Sigma }(2,7)$ and $\mathrm{\Sigma }(3,6)$. In the special case $k=2$ the fan $\mathrm{\Sigma }(2,n)$ is closely related to the metric fan $(n)$, which forms a natural parameter space for the metric spaces on $n$ points. So our results yield a classification of the finite metric spaces on seven points.