The Checkered Smocked Space and its Tangent Cone

TL;DR

This paper explicitly computes the tangent cone at infinity for the checkered smocked space, a specific example of smocked metric spaces, expanding understanding of their geometric structure.

Contribution

It provides an explicit calculation of the norm approximating the pseudometric and determines the tangent cone at infinity for the checkered smocked space.

Findings

The tangent cone at infinity is a normed vector space.

An explicit norm approximating the pseudometric is derived.

The results extend the understanding of smocked metric spaces' asymptotic geometry.

Abstract

Smocked metric spaces were first defined in arXiv:1906.03403 and it was proved that if a norm on the Euclidean space uniformly estimates the pseudometric of a smocked metric space, then the tangent cone at infinity is unique and is a norm vector space with that estimating norm. In this paper, we explicitly calculate the norm approximating the pseudometric of the checkered smocked space and find the tangent cone at infinity.