Assouad–Nagata dimension and gap for ordered metric spaces
Anna Erschler
CNRS, École Normale Superieur, PSL Research University, Paris, FranceIvan Mitrofanov
CNRS, École Normale Superieur, PSL Research University, Paris, France
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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.
Cite this article
Anna Erschler, Ivan Mitrofanov, Assouad–Nagata dimension and gap for ordered metric spaces. Comment. Math. Helv. 98 (2023), no. 2, pp. 217–260
DOI 10.4171/CMH/549