Gromov-Hausdorff convergence and tangent cones of smocked spaces

TL;DR

This paper develops a geometric framework for smocked spaces, showing their convergence properties, tangent cones, and stability under measured Gromov-Hausdorff convergence, thus revealing their controlled limit behavior.

Contribution

It introduces convergence and precompactness results for smocked spaces and constructs examples with arbitrary finite-dimensional normed spaces as tangent cones.

Findings

Smocked spaces converge under Hausdorff and Gromov-Hausdorff conditions.

Finite-dimensional normed spaces can be tangent cones of smocked spaces.

Convergence theory extends to smocked metric measure spaces.

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.