Assouad–Nagata dimension and gap for ordered metric spaces

Abstract

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