Analytic and plurisubharmonic functions in finite and by Michel Herve

By Michel Herve

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Example text

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.

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