Proposition [lcc-001K]

The category of minmax problems has products, given by (X,Y,L) \times (X',Y',L') = (X \times X', Y \oplus Y', (L \times L')), (L \times L')(x,x'; \alpha y + \beta y') = \alpha L(x,y) + \beta L'(x',y')

(L \times L')^+(x,x') = \max (L^+(x),L'^+(x'))

Here Y \oplus Y' denotes the coproduct of convex spaces, and \alpha y + \beta y' is a generic element (note that \alpha ,\beta \in [0,1], \alpha + \beta = 1)

By duality (with (-)^*), it also has coproducts given by (X,Y,L) \oplus (X',Y',L') = (X \oplus X', Y \times Y', L(\alpha x + \beta x';y,y') = \alpha L(x,y) + \beta L'(x',y')).