  We demonstrate that warped Minkowski space backgrounds,
$\mathbb{R}^{n-1,1}\times_w M^{d-n}$, $n\geq3$, that preserve strictly more
than 16 supersymmetries in $d=11$ and type II $d=10$ supergravities and with
fields which may not be smooth everywhere are locally isometric to the
$\mathbb{R}^{d-1,1}$ Minkowski vacuum. In particular, all such flux
compactification vacua of these theories have the same local geometry as the
maximally supersymmetric vacuum $\mathbb{R}^{n-1,1}\times T^{d-n}$.
