Distance problems for typical norms

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I will describe a recent joint work with Matia Bucic and Lisa Sauermann
about the investigation of extremal problems in discrete geometry
for typical norms.
The results include surprisingly tight solutions of the unit and distinct
distances problems for typical norms, as well as a determination of the
chromatic number of the unit distance graph for a typical planar norm.
This settles, in a strong form, questions and conjectures of Matousek,
of Brass, of Chilakamarri and Robertson and of Brass, Moser and Pach.

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