  A remarkable result by S. Artstein-Avidan and V. Milman states that, up to
pre-composition with affine operators, addition of affine functionals, and
multiplication by positive scalars, the only fully order preserving mapping
acting on the class of lower semicontinuous proper convex functions defined on
$\mathbb{R}^n$ is the identity operator, and the only fully order reversing one
acting on the same set is the Fenchel conjugation. Here fully order preserving
(reversing) mappings are understood to be those which preserve (reverse) the
pointwise order among convex functions, are invertible, and such that their
inverses also preserve (reverse) such order. In this paper we establish a
suitable extension of these results to order preserving and order reversing
operators acting on the class of lower semicontinous proper convex functions
defined on arbitrary infinite dimensional Banach spaces.
