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 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.

27 pages. This paper appeared originally as the second part of first versions of arXiv:2011.01732. Now the paper is split it two parts, the first one "Spaces that can be ordered effectively: virtually free groups and hyperbolicity" arXiv:2011.01732, and the second part here