Quantitative Steiner/Schwarz-type symmetrizations

We establish some new quantitative results on Steiner/Schwarz-type symmetrizations, continuing the line of results from [Bourgain et al. (Lecture Notes in Math. 1376 (1988), 44–66)] on Steiner symmetrizations. We show that if we symmetrize high-dimensional sections of convex bodies, then very few steps are required to bring such a body close to a Euclidean ball.