Erdős distance problems in normed spaces

AbstractWe study the problems of the maximum numbers of unit distances, largest distances and smallest distances among n points in a two-dimensional normed space. We determine the exact maximum numbers of smallest and largest distances for each normed space, the maximum number of unit distances for each normed space in which the unit sphere is not strictly convex, and show that the best known upper bound for the euclidean case applies also for each normed space with strictly convex unit sphere, thereby partially answering a question of Erdős and Ulam. The results on smallest distances give also the exact maximum number of touching pairs among n translates of a convex set in the plane, thereby generalizing the results on the translative kissing number by Hadwiger and Grünbaum