  It is shown that for any sufficiently regular even Minkowski valuation $\Phi$
which is homogeneous and intertwines rigid motions, there exists a neighborhood
of the unit ball, where balls are the only solutions to the fixed-point problem
$\Phi^2 K = \alpha K$. This significantly generalizes results by Ivaki for
projection bodies and suggests, via the Lutwak--Schneider class reduction
technique, a new approach to Petty's conjectured projection inequality.
