Subdivisions of Hypersimplices: with a View Toward Finite Metric Spaces

Abstract

The secondary fan $\Sigma (k,n)$ is a polyhedral fan which stratifies the regular subdivisions of the hypersimplices $\Delta (k,n)$. We find new infinite families of rays of $\Sigma (k,n)$, and we compute the fans $\Sigma (2,7)$ and $\Sigma (3,6)$. In the special case $k=2$ the fan $\Sigma (2,n)$ is closely related to the metric fan $\mathrm{MF}(n)$, 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.

21 pages, 2 figures