Given positive integers $d$ and $v$ with $v \geq d+1$, does there always exist a (convex) vertex-transitive $d$-polytope with $v$ vertices? It seems that the answer should be "obviously" true, but I don't know of any natural constructions. A couple of points:
1) If the question is changed to the existence of regular polytopes (i.e., polytopes that are transitive with respect to all the faces), then the answer if 'no' at least for arbitrary $v$.
2) For $d = 2$ the answer is trivially 'yes' by taking the convex hull of $v$ equi-spaced points on the circle.