Definition \Conv and \Conc [lcc-001R]

Let \mathsf {Conv} be the category where objects are pairs (X,f: X \to \mathbb {R}) consisting of a convex space and a convex function, and where morphisms \phi : (X,f) \to (Y,g) are affine maps so that g(\phi (x)) \leq f(x).

Let \mathsf {Conc} be the category where objects are pairs (X,f: X \to \mathbb {R}) consisting of a convex space and a concave function, and where morphisms \phi : (X,f) \to (Y,g) are affine maps so that g(\phi (x)) \geq f(x).