Each convex polytope $P$ has a combinatorial type, its so-called face lattice. This lattice is just the poset of all faces of $P$ ordered by inclusion. Given one realization of such a combinatorial type, one can easily get many others. Just apply an arbitrary projective map to the given realization and the image will be combinatorially equivalent.
Now it would be very nice if one could realize every combinatorial type with vertices on the sphere. Unfortunately, this is not possible. For example, consider the octahedron with pyramids stacked on each facet -- this cannot have all vertices on the sphere (and still be convex).
So my questions is:
Is there a convex body in $\mathbb{R}^d$ such that every possible combinatorial type of a $d$-dimensional convex polytope can be realized with vertices on its surface?
I asked this question on MathOverflow and Sergei Ivanov gave an answer in the affirmative. See here for his beautiful argument.