Fenchel–Young gap ($\Phi (x)+{\Phi }^{\ast }(y)-x\cdot y$)

For a proper convex function $\Phi$ and its convex conjugate ${\Phi }^{\ast }$, 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 $u=|x{|}^2/2-\Phi$, $v=|y{|}^2/2-{\Phi }^{\ast }$.