  For Euclidean space ($\ell_2$), there exists the powerful dimension reduction
transform of Johnson and Lindenstrauss, with a host of known applications.
Here, we consider the problem of dimension reduction for all $\ell_p$ spaces $1
\le p \le 2$. Although strong lower bounds are known for dimension reduction in
$\ell_1$, Ostrovsky and Rabani successfully circumvented these by presenting an
$\ell_1$ embedding that maintains fidelity in only a bounded distance range,
with applications to clustering and nearest neighbor search. However, their
embedding techniques are specific to $\ell_1$ and do not naturally extend to
other norms.
  In this paper, we apply a range of advanced techniques and produce bounded
range dimension reduction embeddings for all of $1 \le p \le 2$, thereby
demonstrating that the approach initiated by Ostrovsky and Rabani for $\ell_1$
can be extended to a much more general framework. We also obtain improved
bounds in terms of the intrinsic dimensionality. As a result we achieve
improved bounds for proximity problems including snowflake embeddings and
clustering.
