The number of unit distances is almost linear for most norms

Details The number of unit distances is almost linear for most norms The number of unit distances is almost linear for most norms

2010-07-07

arxiv.org/abs/1007.1095v1

Mathematics

Combinatorics

Scientific paper

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).

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