On duality and endomorphisms of lattices of closed convex sets

The class of closed convex sets in ℝⁿ is a lattice with respect to the operations of intersections and closed convex hulls of unions. We completely classify the endomorphisms of this lattice and its sublattice consisting of sets containing the origin 0. We show that they consist of constant maps and maps induced by the affine group. As a consequence, we obtain a characterization of the duality mapping for the class of closed convex sets containing 0. Moreover, by modifying a proof of Böröczky and Schneider we obtain the same result for the sublattice consisting of compact bodies containing 0.