# Polyhedral Cones

POLYHEDRAL CONES. *Metasequoia glyptostroboides*, Dawn redwood.

White Plaza.

A polyhedral cone is a shape in space that passes through the origin and that is generated by the solution to a set of linear inequalities. The neighbor-joining algorithm for inference of evolutionary trees involves sequentially identifying pairs of taxa that have the minimal value of a certain linear function of entries in a distance matrix. For a fixed set of n taxa, the possible topologies for their evolutionary tree represent a discrete set. For n taxa, the possible distance matrices represent sets of \(n(n-1)/2\) nonnegative values that map onto the discrete set of possible topologies. As discussed by Eickmeyer et al. (2008), the set of distance matrices that generate a particular output topology according to neighbor-joining can be described as a polyhedral cone.