# Smocked Metric Spaces and their Tangent Cones

## TL;DR

This paper introduces smocked metric spaces, analyzes their geometric properties, and establishes the existence and uniqueness of tangent cones at infinity, which are shown to be normed spaces.

## Contribution

It defines smocked metric spaces, studies their geodesics and limits, and proves tangent cones at infinity are unique normed spaces, advancing understanding of their asymptotic geometry.

## Key findings

- Tangent cones at infinity exist and are unique.
- Tangent cones are normed spaces.
- Analysis of balls and geodesics in smocked spaces.

## Abstract

We introduce the notion of a smocked metric spaces and explore the balls and geodesics in a collection of different smocked spaces. We find their rescaled Gromov-Hausdorff limits and prove these tangent cones at infinity exist, are unique, and are normed spaces. We close with a variety of open questions suitable for advanced undergraduates, masters students, and doctoral students.