  We prove a sharp bilinear inequality for the Klein-Gordon equation on
$\sr^{d+1}$, for any $d \geq 2$. This extends work of Ozawa-Rogers and
Quilodr\'an for the Klein-Gordon equation and generalises work of Bez-Rogers
for the wave equation. As a consequence we obtain a sharp Strichartz estimate
for the solution of the Klein-Gordon equation in five spatial dimensions for
data belonging to $H^1$. We show that maximisers for this estimate do not exist
and that any maximising sequence of initial data concentrates at spatial
infinity.
