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Convex analysis studies convex sets and convex functions, including epigraphs, convex conjugates and subdifferentials.

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.

An infimal convolution is the cheapest way to split a prescribed total between two costs. Suppose two activities contribute $u$ and $v$ to a required total $x$, costing $f(u)$ and $g(v)$. Their combined cost is optimized over all splits $u+v=x$, rather than charging both activities for the entire total.

For proper extended-real functions $f,g:X\to \mathbb{R}\cup \{+\infty \}$ on a real vector space $X$, defineThe definition does not require convexity. An infinite cost forbids that contribution, as with an indicator functional of a constraint set. If no split has two finite costs, set $h(x)=+\infty$; if the costs of admissible splits are unbounded below, $h(x)=-\infty$. Thus proper inputs need not produce a proper extended-real function. The notation $E\square F$ denotes the same operation. See Parikh and Boyd, Proximal Algorithms, §3.1 for the standard convex analysis definition and its connection to the Moreau envelope.

An infimum is a best lower bound, whereas a minimum is a value realized by some split. Writing $\min$ in place of $\inf$ requires an attaining pair $(u_{\ast },v_{\ast })$. For convex functions, mixing two nearly optimal splits and applying convexity to each cost proves that their infimal convolution is convex whenever it is proper; one must use nearly optimal splits because exact minimizers need not exist. Its convex conjugate is $f^{\ast }+g^{\ast }$ whenever the extended real number arithmetic is well defined; see conjugate of an infimal convolution for the precise identity. This turns the optimization over splits into addition in the conjugate variables.

Two quadratic costs: share the load according to its price. Let $X=\mathbb{R}$ and let the convex quadratic functions beFor fixed $x$, eliminate $v=x-u$. The derivative of the total cost is $au-b(x-u)$, so equal marginal costs giveThe cheaper activity receives more of a positive total: when $a<b$ and $x>0$, $u_{\ast }>v_{\ast }$. Completing the square gives an exact certificate for every competing split:Consequently the infimum is a unique minimum, andThe effective quadratic coefficient is $c=ab/(a+b)$, with $1/c=1/a+1/b$. It is smaller than either $a$ or $b$: allowing both activities reduces the cost compared with assigning everything to either one. If $a=b$, each receives half the total and $c=a/2$. This is a simple quadratic optimization problem whose answer preserves the form of the original costs.

Smoothing a corner by splitting its cost. Take the absolute value function $f(u)=|u|$ and the convex quadratic function $g(v)=v^2/(2t)$, with $t>0$. This infimal convolution is the Moreau envelope of the absolute value function:Here $\mathrm{sgn}$ is the sign function, with $\mathrm{sgn}(0)=0$. The optimal $u_{\ast }$ is soft thresholding. In the central region, the quadratic function carries the whole total; outside it, $v_{\ast }$ stays at $t\mathrm{sgn}(x)$ and the absolute value function carries the excess. At $t=1$ the result is the Huber loss in the normalization $x^2/2$ near zero and $|x|-1/2$ outside. Its derivative is $x$ for $|x|\le 1$ and $\mathrm{sgn}(x)$ outside, so the corner becomes a continuous change of slope. Parikh and Boyd's ^[ref] uses twice this normalization for the Huber loss; the explicit formula here fixes the convention. This is also the one-dimensional case of the existing Moreau envelope of the Euclidean norm.

For a functional $E$ on a real Banach space, its convex conjugate is defined on the dual space byHere $⟨p,u⟩$ is the dual pairing. For complex spaces, use its real part, equivalently regard the space as real for convex analysis. The convex conjugate is a convex function, being the supremum of affine functions of $p$, even when $E$ itself is not convex.

Variational analysis studies optimization, perturbation stability and generalized differentiation through functions, sets and set-valued mappings. Convex analysis is a foundational special case; limiting normal constructions also handle nonconvex geometry.