Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Thursday, May 27, 2004, 12:15 pm

Duration: This information is not available in the database

Location: This information is not available in the database

Speaker: Pavel Valtr (Charles Univ., Prague)

Strictly convex norms allowing many unit distances

We construct a (simply defined) strictly convex norm ||.|| in the plane such that for each n there is a set of n points determining Ω(n^4/3) unit distances with respect to ||.||. This gives another evidence that a new proof technique will be needed to improve the currently best known upper bound O(n^4/3) for Erdös' unit-distance problem in the euclidean plane. Our result is asymptotically best possible and disproves a conjecture of Brass. An analogous construction in R³ gives an almost tight lower bound Ω(n^3/2). A connection to lattice problems and to geometric touching problems is also given.

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