Iterations of Minkowski Valuations

TL;DR

This paper proves that iterating certain symmetric Minkowski valuations on convex bodies near the unit ball results in convergence to the sphere, revealing stability properties of these geometric transformations.

Contribution

It establishes convergence of iterated Minkowski valuations to the unit ball for bodies close to it, extending understanding of valuation dynamics in convex geometry.

Findings

Iterated valuations converge to the unit ball in Hausdorff metric.

Convergence applies to bodies in a smooth neighborhood of the sphere.

Results hold for valuations that are homogeneous and rigid motion intertwining.

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.