The reverse Cauchy-Schwarz inequality: for any two causal vectors (orientation in this `squared' formulation is not relevant) $v, w$ we have
$$
\vert g(v,w)\vert^2\ge g(v,v) g(w,w)
$$
is false in pseudo-Riemannian spaces of signature $(p,q)$, $p,q\ge 2$. The proof uses a certain convexity of the unit sphere (not its compactness), and this property holds only for the Lorentzian signature.