Fixed points of Minkowski valuations

It is shown that for any sufficiently regular even Minkowski valuation Φ which homogeneous and intertwines rigid motions, there exists a neighborhood of the unit ball, where balls are only solutions to fixed-point problem Φ2K=αK. This significantly generalizes results by Ivaki projection bodies suggests, via Lutwak–Schneider class reduction technique, new approach Petty's conjectured inequality.