  We disprove a conjecture of Foschi, regarding extremizers for the Strichartz
inequality with data in the Sobolev space
$\dot{H}^{1/2}\times\dot{H}^{-1/2}(\mathbb R^d)$, for even $d\ge 2$. On the
other hand, we provide evidence to support the conjecture in odd dimensions,
and refine his sharp inequality in $\mathbb R^{1+3}$, adding a term
proportional to the distance of the initial data from the set of extremizers.
The proofs use the conformal compactification of the Minkowski space-time given
by the Penrose transform.
