Assouad-Nagata dimension and gap for ordered metric spaces

TL;DR

This paper explores the relationship between Assouad-Nagata dimension and the ordering of metric spaces, providing conditions for optimal orders in TSP and conjectures for characterizing finite AN-dimension spaces.

Contribution

It establishes that finite Assouad-Nagata dimension spaces admit efficient orders for TSP and proposes a conjecture linking AN-dimension to order efficiency in metric spaces.

Findings

Spaces with finite AN-dimension admit good orders for TSP

Sufficient conditions are identified for the converse to hold

Under doubling condition, a stronger gap phenomenon is proven

Abstract

We prove that all spaces of finite Assouad-Nagata dimension admit a good order for Travelling Salesman Problem, and provide sufficient conditions under which the converse is true. We formulate a conjectural characterisation of spaces of finite $AN$-dimension, which would yield a gap statement for the efficiency of orders on metric spaces. Under assumption of doubling, we prove a stronger gap phenomenon about all orders on a given metric space.