  We present examples of metric spaces that are not Riemannian manifolds nor
dimensionally homogeneous that satisfy the Tetrahedral Property. In spite of
that, Euclidean cones over metric spaces with small diameter do not satisfy
this property. We extend Sormani's Tetrahedral Property to a less restrictive
property and prove that this generalized definition retains all the results of
the original Tetrahedral Property proven by Portegies-Sormani: it provides a
lower bound on the sliced filling volume and a lower bound on the volumes of
balls. Thus sequences with uniform bounds on this Generalized Tetrahedral
Property also have subsequences which converge in both the Gromov-Hausdorff and
Sormani-Wenger Intrinsic Flat sense to the same non-collapsed and countably
rectifiable limit space.
