Fenchel–Young gap ($G_f(x,p)=f(x)+f^{\ast }(p)-p\cdot x$)

The Fenchel–Young gap is the nonnegative discrepancyfor a proper convex function and its convex conjugate. It can be $+\infty$. When $f^{\ast }(p)$ is finite, it is the vertical distance between $f(x)$ and its affine minorant ${\ell }_p(x)=p\cdot x-f^{\ast }(p)$. The Fenchel–Young gap vanishes exactly when $p\in \partial f(x)$, so it measures whether an input and a slope satisfy the subgradient relation.

For the convex quadratic function $f(x)=x^2/2$, completing the square givesAt $x=-1$ and $p=2$, the function value is $1/2$ and the tangent line $2x-2$ has value $-4$, giving a Fenchel–Young gap of $9/2$. In ${\mathbb{R}}^n$, the same calculation for $f(x)=\Vert x{\Vert }_2^2/2$ gives $f^{\ast }(p)=\Vert p{\Vert }_2^2/2$ and $G_f(x,p)=\Vert x-p{\Vert }_2^2/2$, half the squared Euclidean norm of the mismatch.

An exact optimization error certificate. Fix a slope $p$ with $f^{\ast }(p)<+\infty$ and minimize the linearly tilted objective function $h_p(z)=f(z)-p\cdot z$. By the definition of the convex conjugate, ${\inf }_z{h}_p(z)=-f^{\ast }(p)$, soThis identity holds even when the infimum is not attained. Thus a computed finite Fenchel–Young gap is exactly the error in objective value for this minimization problem. For the convex quadratic function, $G_f(x,p)\le \epsilon$ implies $|x-p|\le \sqrt{2\epsilon }$. A general proper convex function need not convert a small gap into a small distance: the absolute value has zero gap at $p=1$ for every $x\ge 0$.

The Fenchel–Young gap also connects convex analysis to optimal transport. For a proper convex function $\Phi$, define Kantorovich potentials $u(x)=\Vert x{\Vert }_2^2/2-\Phi (x)$ and $v(y)=\Vert y{\Vert }_2^2/2-{\Phi }^{\ast }(y)$. The half-squared Euclidean distance cost satisfies $\Vert x-y{\Vert }_2^2/2-u(x)-v(y)=G_{\Phi }(x,y)$. For a transport plan with finite integrated cost and integrable Kantorovich potentials, integrating this identity gives the difference between transport cost and the potentials' bound. This is the same local subgradient mismatch summed over transported pairs.