A generalized tetrahedral property

TL;DR

This paper introduces a generalized version of the Tetrahedral Property applicable to a broader class of metric spaces, extending its utility in geometric analysis and convergence studies.

Contribution

It extends Sormani's Tetrahedral Property to a less restrictive form, preserving key geometric bounds and convergence results for metric space sequences.

Findings

Generalized property applies to non-homogeneous metric spaces.

Euclidean cones over small-diameter spaces do not satisfy the property.

Sequences with bounds on the generalized property converge in Gromov-Hausdorff and Intrinsic Flat senses.

Abstract

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.