A sharp bilinear estimate for the Klein-Gordon equation in arbitrary space-time dimensions

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.