  We study various SDP formulations for {\sc Vertex Cover} by adding different
constraints to the standard formulation. We show that {\sc Vertex Cover} cannot
be approximated better than $2-o(1)$ even when we add the so called pentagonal
inequality constraints to the standard SDP formulation, en route answering an
open question of Karakostas~\cite{Karakostas}. We further show the surprising
fact that by strengthening the SDP with the (intractable) requirement that the
metric interpretation of the solution is an $\ell_1$ metric, we get an exact
relaxation (integrality gap is 1), and on the other hand if the solution is
arbitrarily close to being $\ell_1$ embeddable, the integrality gap may be as
big as $2-o(1)$. Finally, inspired by the above findings, we use ideas from the
integrality gap construction of Charikar \cite{Char02} to provide a family of
simple examples for negative type metrics that cannot be embedded into $\ell_1$
with distortion better than $8/7-\eps$. To this end we prove a new
isoperimetric inequality for the hypercube.
