BiConjugate

BiConjugate, or biconjugate, is a concept in convex analysis describing the second Fenchel transform of a function. Let X be a real vector space with dual X*. For a function f: X → (-∞, +∞], its Fenchel conjugate f* is defined by f*(y) = sup_{x∈X} (⟨x, y⟩ − f(x)). The biconjugate is then f**(x) = sup_{y∈X*} (⟨x, y⟩ − f*(y)).

Key properties include: f** is always a proper lower semicontinuous convex function. For all x in X, f**(x) ≤ f(x). If f is proper, lower semicontinuous, and convex, then f** = f. The biconjugate thus acts as the greatest lower semicontinuous convex function that lies below f.

The Fenchel–Moreau theorem states that on a locally convex topological vector space, any proper lower semicontinuous convex function equals its biconjugate. Consequently, f** can be viewed as the closed convex envelope of f: it is the best convex, lsc approximation from below.

Examples illustrate its meaning: if f is the indicator function δ_C of a set C (finite value on C, +∞ otherwise), then f** equals δ_{cl conv(C)}, the indicator of the closure of the convex hull of C. In general, f** provides a convex, lsc surrogate that preserves f only when f is already convex and closed.

Applications of the biconjugate appear in duality theory, optimization, and variational analysis, where it helps relate primal problems to dual formulations and characterizes optimality via subgradients and dual pairs.