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Published online by Cambridge University Press: 17 March 2022
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.
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