Subdivisions of Hypersimplices: With a View Toward Finite Metric Spaces

Abstract

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