= Lipschitz regularization of transport potentials
{title2=$h(x)=\inf_y\{d(x,y)-g(y)\}$}
For <continuous functions> $f,g$ on a <compact metric space> satisfying $f(x)+g(y)\leq d(x,y)$, set $h(x)=\min_y(d(x,y)-g(y))$. The <triangle inequality> makes $h$ $1$-<Lipschitz>, while the constraint gives $f\leq h$ and evaluation at $y=x$ gives $g\leq-h$. Thus the feasible pair $(h,-h)$ has at least as large an objective against any two <probability measures>. This explains why <metric> <optimal transport> duality reduces to a single $1$-<Lipschitz function>.
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