Abstract
We present examples of metric spaces that are not Riemannian manifolds nor dimensionally homogeneous that satisfy Sormani’s Tetrahedral Property. We then note that Euclidean cones over metric spaces with small diameter do not satisfy this property. Therefore, we extend the tetrahedral property to a less restrictive one 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 noncollapsed and countably rectifiable limit space.
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Núñez-Zimbrón, J., Perales, R. A generalized tetrahedral property. Math. Z. 298, 747–769 (2021). https://doi.org/10.1007/s00209-020-02602-9
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DOI: https://doi.org/10.1007/s00209-020-02602-9