Fenchel conjugacy and biconjugation

For a proper extended-real function f on $\mathbb R^d$, define f* at y as the supremum of $\langle y,x\rangle-f(x)$ over finite-domain x. This is the smallest intercept cost making $\langle y,x\rangle-f^*(y)$ a global affine lower bound. Some slopes have infinite cost and provide no finite lower bound. Since a conjugate is a supremum of continuous affine functions of y, it is convex and lower semicontinuous regardless of whether the original f is convex. A conjugate can fail to be proper for a badly behaved original function.

For f(x)=x²/2, completing the square makes the supremum y²/2, attained at x=y. For f(x)=|x|, slopes in [−1,1] have supremum zero, while a slope of magnitude greater than one makes the expression grow without bound on a suitable ray. Hence its conjugate is the indicator of [−1,1]. The conjugate of an indicator is the support function of its set. These examples show why an infinite conjugate value often encodes a geometric slope restriction rather than a numerical difficulty.

The supremum definition immediately gives Fenchel–Young inequality: f(x)+f*(y) is at least the inner product of x and y whenever the sum is meaningful. Equality holds exactly when y is a subgradient of f at x. To prove this equivalence, rewrite the subgradient inequality as an upper bound on $\langle y,z\rangle-f(z)$ and observe that z=x attains it. Conversely, equality of the supremum gives that same upper bound for every z. This converts an apparently abstract conjugate calculation into a checkable optimality relation.

For closed proper convex f, the equality condition is symmetric: y belongs to the subdifferential at x if and only if x belongs to the subdifferential of f* at y. Differentiability is unnecessary, so corners and flat pieces exchange naturally. A corner with an interval of slopes becomes a flat region in the conjugate. If both sides are differentiable and their gradients are mutually invertible on the relevant domains, this relation reduces to inverse gradients, but that stronger smooth picture must not be assumed for every convex function.

The biconjugate is the supremum of all affine lower bounds indexed by y. Therefore it never exceeds f. The finite-dimensional Fenchel–Moreau theorem states that a proper convex lower semicontinuous f equals its biconjugate everywhere, including outside its effective domain. The converse reconstruction uses separation of points below the closed convex epigraph; this is the geometric reason closedness and convexity appear. For a proper convex function that is not lower semicontinuous, biconjugation instead produces its lower semicontinuous closure. This theorem is stated here; the earlier separation argument explains its mechanism rather than serving as a complete proof of every epigraph boundary case.

As a diagnostic, take the indicator of the open interval (−1,1). Its conjugate is |y|, since the endpoint supremum is approached even though not attained. Conjugating again gives the indicator of [−1,1]. Thus the operation retains all affine lower information but cannot remember missing endpoints invisible to suprema. In applications, conjugacy often turns an awkward nonsmooth loss into a simple constraint. The benefit depends on tracking domains carefully: completing a square without checking whether the maximizing point is allowed can give the wrong conjugate formula.