TITLE: Basic question about polytope duals
QUESTION [6 upvotes]: The following must be well known.  Is there a beginning or midlevel
text where the answer is discussed?  Thanks.
Along with a polytope one has the notion of its dual which is officially
defined via the inner product.  However, in three dimensions at least,
the dual is often pictured simply by placing a point in each face and then 
taking the convex hull.  Will this same method work in general?
Question: Let P be an n-dimensional polytope.  Place a point at the barycenter
of each facet of P and designate by Q the convex hull of these points.  Is the
resulting polytope Q combinatorially equivalent to the dual of P ?

REPLY [3 votes]: While the answer to the question of OP is "no", I would like to mention a question raised by Grünbaum and Shephard in "Some problems on polyhedra.", J. Geom. 29 (1987), no. 2, 182–190, for the information of those who may be interested.
They say that a polytope $P$ is of DV-type if it has a combinatorially equivalent realization such that every vertex $v$ of $P$ is an interior point of the face of a dual $P^*$ that corresponds to $v$ in the duality.  The questions
1. Is every polyhedron of DV-type?
2. Does every polyhedorn of DV-type have a dual of DV-type?
3. Which polyhedra of DV-type are also of SV-type? (inscribable w.r.t. a sphere)

I'm not sure about the current status of the question.