  A major open problem in the field of metric embedding is the existence of
dimension reduction for $n$-point subsets of Euclidean space, such that both
distortion and dimension depend only on the {\em doubling constant} of the
pointset, and not on its cardinality. In this paper, we negate this possibility
for $\ell_p$ spaces with $p>2$. In particular, we introduce an $n$-point subset
of $\ell_p$ with doubling constant O(1), and demonstrate that any embedding of
the set into $\ell_p^d$ with distortion $D$ must have
$D\ge\Omega\left(\left(\frac{c\log
n}{d}\right)^{\frac{1}{2}-\frac{1}{p}}\right)$.
