Is the following true? I have a convex polytope $P$ that is vertex transitive - roughly speaking all extremal points of $P$ have the same face-sets (the polytopes are isogonal figures). It is known that all convex polytopes have at least minus faces, where minus-faces are faces with dimension $\mathsf{dim}(P)-1$. Are the extremal points of the dual polytope $P^*$ all correspond to linear functionals that are supporting hyperplanes of the minus-faces of $P$?
To give an intuition for this, if you take a vertex transitive polytope in 3 dimensions (for example a Platonic solid), the extremal points of the dual polytope correspond to the 2-dimensional faces. I want to know if this is true in general for vertex transitive polytopes.