[lcc-001O]

Given a convex space X and a convex function p: X \to \mathbb {R}, we may of course form the convex subspace \{x \mid p(x) \leq 0\}.

Given some minmax problem involving X, L: X \times Y \to \mathbb {R}, we can instead form L' : X \times (Y \times \mathbb {R}^+) \to RR given by L'(x;y,\alpha ) = L(x,y) + \alpha p(x). We have a map into L in this case, given by the identity on X and carrying y to (y,0), which is an alternative "version" of the inclusion of \{x \mid p(x) \leq 0\}.