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1 results

  • 10 - Distinct Distances Variants

  • Adam Sheffer, Bernard M. Baruch College, City University of New York
  • Book:
    Polynomial Methods and Incidence Theory
    Published online:
    17 March 2022
    Print publication:
    24 March 2022, pp 142-154

  • Summary

    After the long and technical proof of the distinct distances theorem, we move to a lighter chapter. In this chapter we study two additional distinct distances problems. We first show that every planar point set contains a large subset that does not span any distance more than once. We then study the structural distinct distances problem: characterizing the point sets that span a small number of distinct distances.

    We also study a problem that does not involve distinct distances, but relies on a variant of Theorem 9.2. This problem considers sets of intervals in the plane that span many trapezoids.