Gromov-Hausdorff convergence and tangent cones of smocked spaces

TL;DR

This paper develops a rigorous convergence theory for smocked spaces, a quotient model obtained by collapsing a collection of disjoint compact sets in Euclidean space. It proves that pointed Gromov-Hausdorff convergence of smocked spaces follows from local Hausdorff convergence of the stitching sets together with uniform local bounds on the smocking constants, and it establishes precompactness under uniform global smocking bounds. It further shows that every finite-dimensional normed space can appear as the tangent cone at infinity of a smocked space, and extends the theory to smocked metric measure spaces with stability under pointed measured Gromov-Hausdorff convergence. These results provide a foundational geometric framework for smocked spaces and suggest avenues for curvature-type analysis and smocked Wasserstein theory.

Abstract

Smocked spaces are a class of metric spaces which were introduced to generalize pulled thread spaces. We investigate convergence of these spaces, showing that if the underlying smocking sets converge in Hausdorff distance and satisfy local uniform bounds on the smocking constants, then the associated smocked spaces converge in the pointed Gromov-Hausdorff sense. We prove a corresponding precompactness result using a similar assumption on the smocking constants. We also show that every finite-dimensional normed vector space arises as the tangent cone at infinity of a suitably constructed smocked space. Finally, we extend the convergence theory to the setting of smocked metric measure spaces, proving stability under pointed measured Gromov-Hausdorff convergence. These results establish a basic geometric framework for smocked spaces and demonstrate that they exhibit controlled limit behavior.