Ovaloids and their symmetry types (Q1207023)

A convex body \(B\) in Euclidean \(n\)-space is called ovaloid if every point of the boundary of \(B\) is an extreme point. The authors show that the set of all ovaloids is dense in the set of all convex bodies (equipped with the Hausdorff metric). Let \(FB\) denote the lattice of all faces of \(B\). The authors prove: If two ovaloids \(B\) and \(C\) have face lattices whose (combinatorial) symmetry groups are isomorphic then \(B\) and \(C\) possess conjugate symmetry groups (subgroups of \(O(n))\) (which is, of course, not true for general convex bodies). Finally, they show that the \(n\)-ball is the only \(n\)-ovaloid with symmetry group \(O(n)\).