On non-diffractive cones

A subject of recent interest in inverse problems is whether a corner must diffract fixed frequency waves. We generalize this question somewhat and study cones $[0,\mathrm{\infty })\times Y$ which do not diffract high frequency waves. We prove that if $Y$ is analytic and does not diffract waves at high frequency then every geodesic on $Y$ is closed with period $2\pi$. Moreover, we show that if $\dim Y=2$, then $Y$ is isometric to either the sphere of radius 1 or its ${\mathbb{Z}}^2$ quotient, $\mathbb{R}{\mathbb{P}}^2$.