LegendreFenchelDualität

The Legendre-Fenchel transform, or convex conjugate, is a central operator in convex analysis. For a function f: R^n → R ∪ {+∞}, it is defined by f*(y) = sup_x {⟨y,x⟩ − f(x)}. The transform generalizes the classical Legendre transform and is well suited to nondifferentiable or non-symmetric settings, as well as functions defined on general vector spaces equipped with a dual pairing.

Key properties include that f* is always convex and lower semicontinuous, even if f is not. Fenchel-Young inequality states that f(x) + f*(y) ≥ ⟨x,y⟩ for all x,y. The biconjugate f**(x) = sup_y {⟨x,y⟩ − f*(y)} provides the convex lower semicontinuous closure of f; for proper lower semicontinuous convex f, one has f = f**. Subgradient relations also hold: y ∈ ∂f(x) if and only if x ∈ ∂f*(y).

In the smooth case, where f is differentiable and strictly convex, the gradient maps are inverses of each other: y = ∇f(x) implies x = ∇f*(y). The transform recovers many familiar forms; for example, if f is quadratic, f*(y) is another quadratic with the inverse matrix. If f is the indicator of a convex set C, then f*(y) is the support function of C.

Applications are widespread in optimization and variational problems, where the transform yields dual formulations and insights into optimality conditions. The Legendre-Fenchel transform thus extends the Legendre transform to a broader, more robust framework for convex analysis and duality.