A lower bound on the distortion of embedding planar metrics into Euclidean space
Ilan Newman, Yuri G. Rabinovich · 2002
(MATH) We exhibit a simple infinite family of series-parallel graphs that cannot be metrically embedded into Euclidean space with distortion smaller than $\Omega(\sqrt\log n\,)$. This matches Rao's general upper bound for metric embedding of planar graphs into Euclidean space, [14], thus resolving the question of how well do planar metrics embed in Euclidean spaces.