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

von a facetFof a simple polytope, is there a projective transformationπsuch that both_{v}π(F)is removable andπ(v)is final inπ(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.