Some characterizing properties of the simplex

We shall prove that a convex body in ℝd (d≥2) is a simplex if, and only if, each of its Steiner symmetrals is a convex double cone over the symmetrization space or, equivalently, has exactly two extreme points outside of this hyperplane. In [3] it is shown that every Steiner symmetral of an arbitrary d-simplex is such a double cone, more precisely a bipyramid. Therefore our main aim is to prove that a convex body which is not a simplex has Steiner symmetrals with more than two extreme points not in the symmetrization space. Some equivalent properties of simplices will also be given.