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.
הערה ביבליוגרפיתFunding Information:
∗This paper is part of the author’s M.Sc thesis, prepared under the supervision of Prof. Shiri Artstein-Avidan. The research was partially supported by ISF grant No. 865/07.