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.

The projection body operator Π, which associates with every convex body in Euclidean space Rn its projection body, is a continuous valuation, it is invariant under translations and equivariant under rotations. It is also well known that Π maps the set of polytopes in Rn into itself. We show that Π is the only non-trivial operator with these properties. MSC 2000: 52B45, 52A20