Convex set questions
In some Hilbert space, let 
be a unit ball polytope of some norm and 
be the unit ball of its dual norm. Is it the case that for every face in there is a vertex of which defines a normal on that face, and vice versa?
I just looked up this stuff, so I barely know what a face is, much less how to tackle this problem right now. My intuition comes only from the knowledge that the dual of the 
ball is the 
ball, in which case it’s easy to see that this is the case.
Another question: what are the vertices of the 
and 
norm balls? (Are these balls even polyhedral? I think so.)
Possibly relevant posts:
- Equivalence of the
norm and the 
norm (1/27/2010) - Trace dual of the
norm (1/19/2010) - An observation on the norming functionals of the
norm ball (2/24/2010)