Distinct distances with ℓp metrics

We study Erdős's distinct distances problem under ℓp metrics with integer p. We prove that, for every ε>0 and n points in R2, there exists a point that spans Ω(n6/7−ε) distinct distances with the other n−1 points. This improves upon the previous best bound of Ω(n4/5). We also characterize the sets that span an asymptotically minimal number of distinct distances under the ℓ1 and ℓ∞ metrics.