LegendreFenchel

The Legendre-Fenchel transform, also known as the convex conjugate, is a fundamental operation in convex analysis that associates to a function f its conjugate f*. For a function f: R^n → R ∪ {+∞} that is proper, convex, and lower semicontinuous, the convex conjugate is defined by f*(y) = sup_x { ⟨x, y⟩ − f(x) }. The biconjugate, f**(x) = sup_y { ⟨x, y⟩ − f*(y) }, satisfies f** = f for all proper closed convex functions, a property that characterizes closed convexity.

Key properties include that f* is convex and lower semicontinuous. Fenchel's inequality states that f(x) + f*(y) ≥ ⟨x, y⟩ for all x, y. The subdifferential relation y ∈ ∂f(x) iff x ∈ ∂f*(y) links the subdifferentials of f and f*. If f is differentiable and strictly convex, the gradient ∇f is invertible, and ∇f*(y) is the inverse mapping of ∇f; equivalently, y ∈ ∂f(x) corresponds to y = ∇f(x).

Special cases and applications: when f is differentiable, the Legendre-Fenchel transform reduces to the classical Legendre transform. The transform is central to duality in optimization, where dual problems often involve f* or its properties. It also appears in thermodynamics, economics, and statistics, and in convex analysis one can recover many classical constructions by applying the conjugate to indicator functions or norm-related functions.

Examples include f(x) = x^2/2 on R, for which f*(y) = y^2/2, and the indicator function of a convex set C, whose conjugate is the support function σ_C(y) = sup_{x∈C} ⟨x, y⟩. See also Fenchel duality, convex conjugate, and subdifferential.

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