Before revision:  Starting with the celebrated Johnson-Lindenstrauss Lemma, such reductions have been studied in depth for the Euclidean (\ell_2) metric and, much less ,  for the Manhattan (\ell_1) metric.
After revision:  Starting with the celebrated Johnson-Lindenstrauss Lemma, such reductions have been studied in depth for the Euclidean (\ell_2) metric and, much less , but much less for the Manhattan (\ell_1) metric.