Iterations of Minkowski Valuations

Abstract

It is shown that for any sufficiently regular even Minkowski valuation \Phi which is homogeneous and intertwines rigid motions, and for any convex body K in a smooth neighborhood of the unit ball, there exists a sequence of positive numbers (\gamma_m)_{m=1}^\infty such that (\gamma_m\Phi^m K)_{m=1}^\infty converges to the unit ball with respect to the Hausdorff metric.