The number of unit distances is almost linear for most norms
Combinatorics
2010-07-08 v1
Abstract
We prove that there exists a norm in the plane under which no n-point set determines more than O(n log n log log n) unit distances. Actually, most norms have this property, in the sense that their complement is a meager set in the metric space of all norms (with the metric given by the Hausdorff distance of the unit balls).
Cite
@article{arxiv.1007.1095,
title = {The number of unit distances is almost linear for most norms},
author = {Jiří Matoušek},
journal= {arXiv preprint arXiv:1007.1095},
year = {2010}
}