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 $[0,infty)times Y$ 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 $2pi$. 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, $mathbb{R}mathbb{P}^2$.
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