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.