  We study Erd\H os's distinct distances problem under $\ell_p$ metrics with
integer $p$. We improve the current best bound for this problem from
$\Omega(n^{4/5})$ to $\Omega(n^{6/7-\epsilon})$, for any $\epsilon>0$. We also
characterize the sets that span an asymptotically minimal number of distinct
distances under the $\ell_1$ and $\ell_\infty$ metrics.
