Affine invariant points and new constructions

In [2] Grünbaum asked if the set of all affine invariant points of a given convex body is equal to the set of all points invariant under every affine automorphism of the body. In [3] we have proven the case of a body with no nontrivial affine automorphisms. After some partial results [6] and [7] the problem was solved in positive by Mordhorst [8]. In this note we provide an alternative proof of the affirmative answer, developing the ideas of [3]. Moreover, our approach allows us to construct a new large class of affine invariant points.