  Employing a local version of the Brunn-Minkowski inequality, we give a new
and simple proof of a result due to Andrews, Choi and Daskalopoulos that the
origin-centred balls are the only closed, self-similar solutions of the Gauss
curvature flow. Extensions to various non-linearities are obtained, assuming
the centroid of the enclosed convex body is at the origin. By applying our
method to the Alexandrov-Fenchel inequality, we also show that origin-centred
balls are the only solutions to a large class of even Christoffel-Minkowski
type problems.
