Dual affine invariant points

An affine invariant point on the class of convex bodies K_n in Rⁿ, endowed with the Hausdorff metric, is a continuous map from K_n to Rⁿ that is invariant under one-to-one affine transformations A on Rn, that is, p(A(K)) = A(p(K)). We define here the new notion of dual affine point q of an affine invariant point p by the formula q(K^p(K)) = p(K) for every K K_n, where K^p(K) denotes the polar of K with respect to p(K). We investigate which affine invariant points do have a dual point, whether this dual point is unique and has itself a dual point. We also define a product on the set of affine invariant points, in relation with duality. Finally, examples are given which exhibit the rich structure of the set of affine invariant points.