Fenchel–Young gap
= Fenchel–Young gap
{c}
{title2=$\Phi(x)+\Phi^*(y)-x\cdot y$}
For a <proper convex function> $\Phi$ and its <convex conjugate> $\Phi^*$, this gap is nonnegative by the <Fenchel–Young inequality>. It vanishes exactly when $y\in\partial\Phi(x)$. Its integral against a <transport plan> is the difference between the half-squared-distance cost and the value of the integrable <Kantorovich potentials> $u=|x|^2/2-\Phi$, $v=|y|^2/2-\Phi^*$.