By Richard. Abbatt

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6) As an upper envelope of convex lsc functions, fˆ is convex and lsc and fˆ(x) ≤ f (x). 3, there exists n ∈ A such that n (y) = µn µn → f (y). As (y, µn ) ∈ and f (x) ≥ n (x) for all x. By going to the limit n (y) ≤ fˆ(y) ≤ f (y) ⇒ f (y) = lim n→∞ n (y) ≤ fˆ(y) ≤ f (y) 15 Cf. R. T. Rockafellar [1, Cor. 1, p. 103] for another proof. 9. Convexiﬁcation and Fenchel–Legendre Transform 49 and fˆ(y) = f (y). Since each ∈ A is of the form (x) = a · x + b, we get fˆ(x) = sup a∈Rn and b∈R ∀y∈Rn , a·y+b≤f (y) a · x + b = sup a · x + inf n [f (y) − a · y] a∈Rn y∈R since ∀y ∈ Rn , a · y + b ≤ f (y) ⇒ b ≤ inf n [f (y) − a · y] = − sup [a · y − f (y)] .

6, it is compact. 1, there exists y ∈ argmin gε/η such that f (y) = ∀x ∈ Rn , f (y) + inf z∈argmin gε/η f (z) ε ε y − xε ≤ f (x) + x − xε . η η For x = xε , f (y) + ε y − xε ≤ f (xε ) η ⇒ f (y) ≤ f (xε ) − ε y − xε η and, from the definition of xε , f (y) + ε y − xε ≤ f (xε ) < inf n f (x) + ε ≤ f (y) + ε x∈R η ⇒ y − xε < η. Consider the new function def g(z) = f (z) + ε z−y . η 9 From the proof of J. M. Borwein and A. S. Lewis [1, Thm. 2, pp. 153–154]. 4) 6. Ekeland’s Variational Principle 31 By definition, inf z∈Rn g(z) ≤ f (y) and ε ε ε z − xε ≤ f (z) + z−y + y − xε ∀z ∈ Rn , f (z) + η η η ε ε z − xε ⇒ f (y) + + y − xε = infn f (z) + z∈R η η ε ε ≤ infn f (z) + z−y + y − xε z∈R η η ε ε ⇒ f (y) ≤ infn f (z) + z − y ≤ f (y) ⇒ f (y) = infn f (z) + z−y z∈R z∈R η η .

Iii) If dom f = ∅, then ri (dom f ) = dom f . Proof. 1(i), dom f is convex. 1(i), applied to aff (dom f ), ri (dom f ) is also convex. (ii) If dom f is a singleton, then ri (dom f ) = dom f = ∅. If the dimension of aff (dom f ) is greater than or equal to one, then, by convexity of dom f , U contains an open nonempty segment that is contained in ri (dom f ). (iii) From part (ii), ∅ = ri (dom f ) ⊂ aff (dom f ). 1(v), ri (dom f ) = dom f . 2. Let f : Rn → R ∪{+∞}, dom f = ∅, be a convex function.

### A treatise on the calculus of variations by Richard. Abbatt

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