Determining classes of convex bodies by restricted sets of Steiner symmetrizations (Q1117462)

Let \(\bar S_{(k)}^{d-2}\) denote the union of k arbitrary, but pairwise-distinct great (d-2)-subspheres of the unit (d-1)-sphere in \({\mathbb{R}}^ d(d\geq 2)\). It is shown that a convex body K in \({\mathbb{R}}^ d\) is a d-simplex if and only if for each direction \(u\in \bar S_{(3)}^{d-2}\) the corresponding Steiner symmetral has precisely two extreme points outside its symmetrization hyperplane. Weaker conditions lead one to characterizations of additional classes of convex bodies, e.g.: If the same holds for each \(u\in \bar S_{(2)}^{d-2}\), then K has to be a d-simplex or a (d-2)-fold pyramid over a planar, convex 4-gon b, whose diagonals \(D_ i\) satisfy \[ lin S_ i^{d-2}\| aff(D_ i\cup (ext K\setminus ext B))\quad with\quad \cup S_ i^{d-2}=\bar S_{(2)}^{d-2}\quad and\quad i=1,2. \]