  In [7], a notion of constant scalar curvature metrics on piecewise flat
manifolds is defined. Such metrics are candidates for canonical metrics on
discrete manifolds. In this paper, we define a class of vertex transitive
metrics on certain triangulations of $\mathbb{S}^3$; namely, the boundary
complexes of cyclic polytopes. We use combinatorial properties of cyclic
polytopes to show that, for any number of vertices, these metrics have constant
scalar curvature.
