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 ?