  In the semi-classical limit, the quantum mechanics of a stationary beam of
counter-streaming relativistic electrons and ions is described by a nonlinear
system of finite-temperature Thomas-Fermi equations. In the high temperature /
low density limit these Thomas-Fermi equations reduce to the (semi-)conformal
system of Bennett equations discussed earlier by Lebowitz and the author. With
the help of a sharp isoperimetric inequality it is shown that any hypothetical
particle density function which is not radially symmetric about and decreasing
away from the beam's axis would violate the virial theorem. Hence, all beams
have the symmetry of the circular cylinder.
