We shall prove that a convex body in R d (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 I-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.
1. NOTATION AND DEFINITIONS
Let ~a(d ~> 2) denote the d-dimensional Euclidean vector space with scalar product (.,.) and unit sphere S d-1 := {u~ ~d: (u,u) = 1}. We write K d for the set of convex bodies, i.e. compact, convex subsets of R d with non-empty interior. For other notation and abbreviations the reader should consult [1] and I-2], respectively.
In particular, let Q be a (d -1)-dimensional compact, convex set and let I be a line segment not contained in aft Q. If Q n I is a single point belonging to Q c~ relint I, then the convex body B: = conv(Q w I) is called a convex double cone over aft Q. (Clearly, B has exactly two extreme points outside aft Q.) Further, for u ~ S d-1 let H = {x ~ ~a: (x, u) = 0} be an arbitrary (d -1)subspace of Na and let G be a line orthogonal to H. If G meets K ~ K a, then let Sn(G c~ K) be the line segment in G and of the same length as G n K, which is centred about G c~ H. The union of all such segments SH(K c~ G) is the Steiner symmetral SH(K ) of K with respect to u.
Under Steiner symmetrization of K e K d two well-known measures of this body (also depending on the direction of symmetrization) are invariant. The first, which is the area of the orthogonal image of K in H, is called the outer (d -1)-quermass or briohtness V'a_I (K,u ). The second measure, namely the length of the longest chord of K in direction u, is called the inner 1-quermass ~I(K, u). It is further known that V d_ I(K, u) is the restriction to S d-1 of the support function of a convex body HK, called the projection body of K, so that F a_ I(K, u)= h(HK, u) = sup{(x, u) : x ~ IlK} for u ~ S a-1, whereas (again for u e S a-1) ~BI(K, u) is the radial function