Kolloquien und Oberseminare Übersicht Detail

Speaker:
Dr. Maria Infusino

Steiner symmetrization is a powerful symmetrization process, which is often used to identify the ball as the solution to geometric optimization problems. It is indeed well-known that sequences of iterated Steiner symmetrizations of any given compact set converge to a ball for most sequences of directions. However, they generally fail to converge along a sequence of directions where the differences between successive angles are square summable. In this lecture we present a result by Bianchi et al. of 2012, showing that such class of sequences actually converge in shape, namely, when the Steiner symmetrizations are followed by suitable rotations. Moreover, we provide explicit examples showing that the limit in this case need not be a convex set. The techniques used in proving this result also allow to establish convergence of iterated Steiner symmetrizations in other situations without the usual convexity assumption on the starting compact set, e.g. along a Kronecker sequence of directions on the unit circle.