Along with a polytope P 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 of P and then taking the convex hull. Will this same method work in general?
Question: Let P be an n-dimensional polytope. Place points 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$\,$?
Thanks.
No. For example, let P be a regular icosahedron with each vertex perturbed randomly by a small amount. Your constructed Q will consist of 36 triangles, rather than 12 pentagons.