Determining classes of convex bodies by restricted sets of Steiner symmetrizations

✍ Scribed by Horst Martini

• Publisher: Springer
• Year: 1989
• Tongue: English
• Weight: 333 KB
• Volume: 30
• Category: Article
• ISSN: 0046-5755
• DOI: 10.1007/bf00181340

✦ Synopsis

In [4]

it was shown that a convex body in R a (d >/2) is a simplex if and only if each of its Steiner symmetrals has exactly two extreme points outside the corresponding symmetrization space. A natural question arises about restricted sets of symmetrization directions which guarantee this characterization of simplices. Let ~2 denote an arbitrary triple of pairwisedistinct great (d -2)-spheres on the unit sphere of R e. We shall prove that a convex body K is a simplex if and only if for every direction u E g~ 2 the corresponding Steiner symmetral of K has the property described above. Weaker conditions characterize additional classes of convex bodies, e.g. (d -2)-fold pyramids over planar, convex 4-gons.