24 1. The Kantorovich Duality Theorem 1.9 (Fenchel-Rockafellar duality). Let E be a normed vector space, E∗ its topological dual space, and Θ, Ξ two convex functions on E with values in R ∪{+∞}. Let Θ∗, Ξ∗ be the Legendre-Fenchel transforms of Θ, Ξ respectively. Assume that there exists z0 E such that Θ(z0) +∞, Ξ(z0) +∞, Θ is continuous at z0. Then, (1.9) inf E Θ + Ξ = max z∗∈E∗ −Θ∗(−z∗) Ξ∗(z∗) . Remark 1.10. It is part of the theorem that the supremum in the right- hand side above is a maximum. As the reader can check, identity (1.9) is really a minimax theorem. The proof given in [64] is a clever but rather easy consequence of the Hahn- Banach theorem of separation of convex sets, a proof of which can be found in [217, 64] or many other sources. We briefly sketch the argument towards Theorem 1.9, so that the reader can understand where the continu- ity assumption comes into play. Proof of Theorem 1.9. What we should prove is sup z∗∈E∗ inf x,y∈E Θ(x) + Ξ(y) + z∗,x y = inf x∈E Θ(x) + Ξ(x) . 1. The choice x = y shows that the left-hand side is not larger than the right-hand side so we only have to prove the existence of a linear form z∗ E∗ such that (1.10) ∀x, y E, Θ(x) + Ξ(y) + z∗,x y m inf(Θ + Ξ). Since Θ(z0) + Ξ(z0) +∞, the infimum m is finite. 2. Let C (x, λ) E × R λ Θ(x) , C (y, μ) E × R μ m Ξ(y) . Since Θ and Ξ are convex, so are C and C . From the assumptions in Theorem 1.9 we deduce that (z0, Θ(z0) + 1) Int(C), and in particular C has nonempty interior this easily implies that C = Int(C). Moreover, C and C are disjoint, because m = inf(Θ + Ξ). It follows from the Hahn-Banach theorem that there exists a nontrivial linear form (E × R)∗ satisfying inf c∈C , c = inf c∈Int(C) , c sup c ∈C , c .
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