Basic question about polytope duals

Question

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 ?

Answer 1

This is false, even in dimension $3$. Take a regular icosahedron. Then wiggle the vertices a bit -- still get an icosahedron, but it is not regular. If you take the barycenters and the convex hull, you will generally only get triangular faces. This is because if you have a vertex $p_0$ and adjacent vertices $p_1,\ldots,p_5$, then the new vertices are, up to a shift and scaling $$p_1+p_2,p_2+p_3,p_3+p_4,p_4+p_5,p_5+p_1.$$ There is no reason for them to lie in the same affine plane. However, the dual is still combinatorially a dodecahedron.

Answer 2

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.

Answer 3

So I wrote a undergrad thesis on polytopes and I used Grunbaum's Convex Poltyopes.

I also found the notes of Alexander Woo really helpful, unfortunately, I can't find his original notes I was using 10 years ago, but he has these notes here. If you want to look at my notes from undergrad, they might be simpler to read if it's what you're looking for, here.