Let P be a simple polytope defined as an intersection of n halfspaces.
A facet F of P, supported by halfspace H, is removable if the intersection of the remaining (n-1) halfspaces is bounded. F is projectively removable if there exists a projective transformation π such that π(F) is removable from π(P).
It is easy to show that every facet of a simple d-polytope with at least (d+2) facets is projectively removable, since there is a projective transformation mapping (d+1) of the remaining halfspaces into a (d)-simplex.
Consider a vertex v defined by the intersection of d halfspaces and lying on a removable facet F supported by halfspace H. Suppose we translate H along its normal axis away from the center of the polytope out towards infinity. As we do so, some vertices will disappear from F as the corresponding halfspaces no longer intersect H within the polytope. At the time just before H leaves the polytope entirely there will be exactly d vertices left on F.
Define v to be a final vertex if, when H is translated out of P in this way, v is one of the d vertices remaining on F.
In a given realization of a polytope, some set of d vertices on a removable facet will be final. But if we apply an appropriate projective transformation, can any vertex v be made into a final vertex? In other words,
for any vertex v on a facet F of a simple polytope, is there a projective transformation πv such that both π(F) is removable and π(v) is final in π(P)?
Based on what I believe I understand about projective transformations, I can imagine that there is a projective transformation that shrinks F to an arbitrarily small point, and another transformation that perturbs F so that a given vertex "sticks out" enough to be a final vertex. However, I am not clear how to show from a formal definition of projective transformations that such transformations always exist or that there exists a given transformation that imposes both properties.
As a continuation of this question, let me ask: how can I gain more intuition about what properties of a polytope can be modified by projective transformation? Can facets be scaled arbitrarily and edges shifted around as I have suggested? I have taken a look at some texts suggested in Ziegler about projective geometry, but I'm interested in knowing more precisely what kind of things projective transformations can and cannot do to polytopes.