On combinatorial and affine automorphisms of polytopes (Q797836)

Given a d-dimensional convex polytope P, is there a polytope Q, combinatorially equivalent to P, such that every combinatorial automorphism of Q is induced by an affine transformation of the surrounding Euclidean space? This long standing question of M. Perles has a positive answer for \(d=3\), and also for all polytopes with few vertices. The present paper shows that the answer is negative, in general. It gives an explicit 4-dimensional counterexample P, with 10 vertices.