Proposition: Finite increment obstruction

Suppose a generated affine representation uses only finitely many edge displacement vectors. Its generated cone is finitely generated. In dimension at least three it is not linearly isomorphic to a Lorentz cone.

Proof. Every reachable displacement is a sum of edge displacements, and every edge displacement is reachable. Thus the two sets generate the same cone. A finitely generated cone is closed and polyhedral; equivalently, intersecting a pointed such cone with a strictly positive affine section gives the convex hull of finitely many normalized generators. It has finitely many extreme rays. A Lorentz cone in dimension at least three has infinitely many, as proved in Proposition prop:orthant.