The Ekeland Variational Principle, the Bishop-Phelps Theorem, and the Brøndsted-Rockafellar Theorem Our aim is to prove the Ekeland Variational Principle which is an abstract result that found numerous applications in various fields of Mathematics. As its application to Convex Analysis, we provide a proof of the famous Bishop- Phelps Theorem and some related results. Let us recall that the epigraph and the hypograph of a function f : X ! [−∞; +1] (where X is a set) are the following subsets of X × R: epi f = f(x; t) 2 X × R : t ≥ f(x)g ; hyp f = f(x; t) 2 X × R : t ≤ f(x)g : Notice that, if f > −∞ on X then epi f 6= ; if and only if f is proper, that is, f is finite at at least one point. 0.1. The Ekeland Variational Principle. In what follows, (X; d) is a metric space. Given λ > 0 and (x; t) 2 X × R, we define the set Kλ(x; t) = f(y; s): s ≤ t − λd(x; y)g = f(y; s): λd(x; y) ≤ t − sg ⊂ X × R: Notice that Kλ(x; t) = hyp[t − λ(x; ·)]. Let us state some useful properties of these sets. Lemma 0.1. (a) Kλ(x; t) is closed and contains (x; t). (b) If (¯x; t¯) 2 Kλ(x; t) then Kλ(¯x; t¯) ⊂ Kλ(x; t). (c) If (xn; tn) ! (x; t) and (xn+1; tn+1) 2 Kλ(xn; tn) (n 2 N), then \ Kλ(x; t) = Kλ(xn; tn) : n2N Proof. (a) is quite easy. To show (b), let (¯x; t¯) 2 Kλ(x; t) and (y; s) 2 Kλ(¯x; t¯). Then λd(x; y) ≤ λd(x; x¯) + λd(¯x; y) ≤ (t − t¯) + (t¯− s) = t − s, and hence (y; s) 2 Kλ(x; t). To prove (c), we shall use (a) and (b). Notice that Kλ(xn+1; tn+1) ⊂ Kλ(xn; tn) for each n. Since (xm; tm) 2 Kλ(xn; tn) whenever m ≥ n, we can pass to the limit for m ! +1 to obtain (x; t) 2 Kλ(xn; tn) for each n. Thus T Kλ(x; t) ⊂ n Kλ(xn; tn). To show the equality, take (y; s) 2= Kλ(x; t), that is, λd(x; y) − t + s > 0. But then λd(xn; y) − tn + s > 0 for each sufficiently large T n. It follows that (y; s) 2= n Kλ(xn; tn). The proof of the following proposition, basic for Ekeland's principle, is some- how \geometric" and its basic idea is quite simple. The reader is invited to draw a picture to grasp this geometric idea of the proof. 1 2 Proposition 0.2. Let (X; d) be a complete metric space, and let PR denote the canonical projection of X × R onto R, that is, PR(x; t) = t. Let F ⊂ X × R be a closed set such that inf PR(F ) > −∞. Then for every λ > 0 and (x0; t0) 2 F there exists (¯x; t¯) 2 F \ Kλ(x0; t0) such that F \ Kλ(¯x; t¯) = f(¯x; t¯)g: P1 Proof. Fix a sequence f"ngn≥1 ⊂ (0; +1) such that 1 "n < +1. Let us inductively define sets Fn and points (xn; tn) in X × R for n ≥ 0. Case n = 0. Define F0 = F , and recall that we already have (x0; t0) 2 F0. Now let n ≥ 0, and assume that we have already defined Fk and (xk; tk) 2 Fk for 0 ≤ k ≤ n. Put Fn+1 = Fn \Kλ(xn; tn), and fix a point (xn+1; tn+1) 2 Fn+1 such that tn+1 ≤ inf PR(Fn+1) + "n+1. Notice that Fn+1 ⊂ Fn and Fn+1 = F \ Kλ(xn; tn)(n ≥ 0) by Lemma 0.1. Moreover, since (xn+1; tn+1) 2 Fn \ Kλ(xn; tn), we have that λd(xn; xn+1) ≤ tn − tn+1 ≤ inf PR(Fn) + "n − inf PR(Fn) = "n (n ≥ 1). Our choice of f"ng easily implies that the sequences fxng and ftng are Cauchy and hence ¯ T T convergent: (xn; tn) ! (¯x; t) 2 F1. Now, n Fn = n[F \ Kλ(xn; tn)] = T ¯ ¯ F \ n Kλ(xn; tn) = F \ Kλ(¯x; t) by Lemma 0.1. To show that F \ Kλ(¯x; t) is a singleton, it suffices to show that diam Fn ! 0. (On X × R we can consider, e.g., the metric d^(x; t); (y; s) := maxd(x; y); jt − sj , or d^(x; t); (y; s) := d(x; y) + jt − sj which is clearly equivalent to the previous one.) For n ≥ 2 and (y; s) 2 Fn = Fn−1 \ Kλ(xn−1; tn−1), we have as above: λd(xn−1; y) ≤ tn−1 − s ≤ inf PR(Fn−1) + "n−1 − inf PR(Fn−1) = "n−1 ! 0. It follows that diam(Fn) ! 0, and we are done. We are ready for the famous \variational principle" due to I. Ekeland [J. Math. Anal. Appl. 47 (1974), 324{353]. It asserts that, under apropriate assumptions, if a point x0 is an \almost minimum point" for a function f then there exists a small Lipschitz perturbation of f attaining its (strict global) minimum at a point \near" to x0. As usual, \l.s.c." stands for \lower semicontinuous". Theorem 0.3 (Ekeland Variational Principle). Let (X; d) be a complete metric space, and f : X ! (−∞; +1] a proper l.s.c. function which is bounded below. Let " > 0, λ > 0, x0 2 X, and f(x0) ≤ inf f(X) + ". Then there exists x¯ 2 X such that (a) f(¯x) ≤ f(x0); (b) d(x0; x¯) ≤ ε/λ; (c) f(¯x) < f(x) + λd(¯x; x) for all x 2 X n fx¯g (that is, the function f + λ(¯x; ·) attains its strict global minimum at x¯.) Proof. The set F := epi f is a nonempty closed set in X × R,(x0; f(x0)) 2 F , f(x0) ≤ inf PR(F ) + " and the infimum is finite. By Proposition 0.2, there is (¯x; t¯) 2 F \ Kλ(x0; f(x0)) such that F \ Kλ(¯x; t¯) = f(¯x; t¯)g. Since (¯x; f(¯x)) 2 F \ Kλ(¯x; t¯), we necessarily have t¯ = f(¯x). Moreover, since (¯x; f(¯x)) 2 3 Kλ(x0; t0), we have λd(x0; x¯) ≤ f(x0) − f(¯x) ≤ inf f(X) + " − inf f(X) ≤ " which implies (a) and (b). Finally, if x 6=x ¯ then (x; f(x)) 2= Kλ(¯x; f(¯x)), that is, λd(¯x; x) > f(¯x) − f(x) which gives (c). p Remark 0.4. Notice that for λ = " , Theorem 0.3(b,c) become p p d(x0; x¯) ≤ " and f(¯x) < f(x) + " d(¯x; x) whenever x 6=x ¯. 0.2. The Bishop-Phelps Theorem. The classical Bishop-Phelps Theorem asserts that if X is a (real) Banach space then the elements of X∗ that attain the norm are dense in X∗. We shall prove a more general version of this theorem, where the unit ball of X is replaced by an arbitrary closed, convex, proper subset of X. We shall use the following standard terminology. If C is a nonempty set in a normed space X, x 2 C, x∗ 2 X∗ n f0g and x∗(x) = sup x∗(C), then we say that x is a support point of C and x∗ is a support functional for C. Notice that each support point belongs to the boundary @C of C, and all strictly positive multiples of a support functional are again support functionals. Moreover, the Hahn-Banach Theorem immediately gives that if C is a closed convex set with nonempty interior (in some topological vector space) then each point of @C is a support point of C. The main tool for the Bishop-Phelps Theorem will be the following auxiliary theorem. In the proof we shall use the easy and well-known fact that a closed hyperplane in X ×R that strictly separates two distinct points having the same first component coincides with the graph of a continuous affine function on X. Theorem 0.5. Let X be a Banach space, C ⊂ X a nonempty, closed, convex ∗ ∗ set. Let " > 0, x0 2 C and x0 2 X n f0g be such that ∗ ∗ ∗ inf x0(C) > −∞ and x0(x0) ≤ inf x0(C) + ": ∗ ∗ ∗ Then for each 0 < λ < kx0k there exist x1 2 C and x1 2 X n f0g such that " kx − x k ≤ ; kx∗ − x∗k ≤ λ and x∗(x ) = inf x∗(C) : 1 0 λ 1 0 1 1 1 Proof. The function ( x∗(x) if x 2 C, f(x) = 0 +1 otherwise, ∗ is a proper, convex, l.s.c. function on X (since epi f = (epi x0) \ C × R), and it satisfies f(x0) ≤ inf f(X) + ". By the Ekeland Variational Principle, there exists x1 2 X such that f(x1) ≤ f(x0), kx1 − x0k ≤ ε/λ, and f(x1) ≤ f(x) + λkx − x1k for all x 2 X. The first inequality gives that x1 2 C, while the last one can be rewritten in the form g(x) := f(x) − f(x1) ≥ −λkx − x1k =: h(x); x 2 X: 4 Since g is proper, convex and l.s.c., h is finite, concave and continuous, and g(x1) = h(x1) = 0, we can separate the epigraph epi g and the hypograph hyp h (which has nonempty interior!) by a closed hyperplane in X × R that ∗ ∗ necessarily passes through the point (x1; 0). Thus there exists y 2 X such that ∗ f(x) − f(x1) ≥ y (x − x1) ≥ −λkx − x1k; x 2 X: ∗ ∗ ∗ ∗ The first inequality can be written as x0(x) − x0(x1) ≥ y (x) − y (x1), x 2 C, that is, ∗ ∗ ∗ ∗ (x0 − y )(x) ≥ (x0 − y )(x1); x 2 C; while the second inequality gives (by putting x = x1 + v) y∗(v) ≤ λkvk; v 2 X; ∗ ∗ ∗ ∗ that is, ky k ≤ λ.