By Michel Herve
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Additional info for Analytic and plurisubharmonic functions in finite and infinite dimensional spaces. Course given at the University of Maryland, Spring 1970
Deduce from this the inequality between the geometric and arithmetic means: √ n a1 + a 2 + · · · + a n n a1 · a 2 · . . · a n (a i > 0) . 30 Example. Integral functionals, which are very important to us, are naturally extended-valued in many cases. Let Λ : [ 0,1] × R × R → R be continuous and bounded below. For x ∈ X = AC[ 0,1], we set f (x) = 1 0 Λ t, x(t), x ′(t) dt. 3 Convex functions 37 Under the given hypotheses, the composite function t → Λ (t, x(t), x ′ (t)) is measurable and bounded below, so that its (Lebesgue) integral is well defined, possibly as +∞.
Prove that co S is compact. 4 admit alternate characterizations, as we now see. 2 (pp. 24–25) that are convex. 9 Proposition. Let S be a convex set in X , and let x ∈ S . Then TS (x) is convex, S ⊂ x + TS (x) , and we have TS (x) = cl u−x : t > 0, u ∈ S , NS (x) = ζ ∈ X ∗ : ⟨ ζ , u − x ⟩ t 0 ∀u ∈ S . Proof. 2) that the following set W is convex: cl u−x : t > 0, u ∈ S . t It is clear from the definition of tangent vector that TS (x) ⊂ W . To prove the opposite inclusion, it suffices to show that any vector of the form v = (u − x)/t , where u is in S and t > 0, belongs to TS (x), since the latter is closed.
39; there remains to prove that this is a sufficient condition for x to be a solution of the optimization problem min A f (when f and A are convex). 3 Convex functions 35 Let u be any point in A. Then v := u − x belongs to TA (x), by Prop. 9, and we have (by Prop. 22) f (u) − f (x) = f (x + v) − f (x) ⟨ f ′ (x), v ⟩. This last term is nonnegative, since − f ′ (x) ∈ NA (x), and since the normal cone is the polar of the tangent cone; it follows that f (u) f (x). ⊓ Criteria for convexity. The following first and second order conditions given in terms of derivatives are useful for recognizing the convexity of a function.