Title: On the Impossibility of Dimension Reduction for Doubling Subsets of $\ell_p$
Authors: Yair Bartal, Lee-Ad Gottlieb and Ofer Neiman
Abstract:
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 $D\ge\Omega\left(\left(\frac{c\log n}{d}\right)^{\frac{1}{2}-\frac{1}{p}}\right)$.