Convergence in shape of Steiner symmetrizations (Q2851030)

The authors study the convergence of iterates of Steiner symmetrals applied to a non-empty compact set \(K\) in \(\mathbb R^n\). The main result is Theorem 2.2 saying that for any sequence \((u_m)\) of norm one vectors in \(\mathbb R^n\) such that \(u_{m-1}\cdot u_m=\cos(\alpha_m)>0\) with \(\sum_{m=1}^\infty \alpha_m^2<\infty\) there is a sequence \((\mathcal R_m)\) of rotations of \(\mathbb R^m\) such that for every non-empty compact set \(K\subset\mathbb R^m\) the rotated iterated Stein symmetrals \(K_m=\mathcal R_m\mathcal S_{u_m}\cdots\mathcal S_{u_1}(K)\) converge to some compact set \(L\) both in Hausdorff distance and in the metric equal to the Lebesgue measure of symmetric difference. Examples 2.3 and 2.4 show that the limit set \(L\) needs not be neither an ellipsoid (for a convex \(K\)), nor a convex set (for an arbitrary \(K\)). In the final sixth section the authors generalize a recent result of Klain proving that for any sequence \((u_m)\) of norm one vectors taken from a finite set \(F=\{u_m\}_{m=1}^\infty\) and for any non-empty compact set \(K\subset\mathbb R^n\) the sequence of iterated Steiner symmetrals \(K_m=\mathcal S_{u_m}\cdots\mathcal S_{u_1}(K)\) converges to some compact set \(L\) which is symmetric with respect to any direction that appears infinitely many times in the sequence \((u_m)\).