By Constantin P. Niculescu, Lars-Erik Persson (auth.)

ISBN-10: 0387243003

ISBN-13: 9780387243009

ISBN-10: 0387310770

ISBN-13: 9780387310770

Convex capabilities play a tremendous function in lots of branches of arithmetic, in addition to different parts of technology and engineering. the current textual content is aimed to a radical creation to modern convex functionality idea, which includes a strong and chic interplay among research and geometry. a wide number of topics are coated, from one genuine variable case (with all its mathematical gem stones) to a couple of the main complicated issues comparable to the convex calculus, Alexandrov’s Hessian, the variational technique of partial differential equations, the Prékopa-Leindler kind inequalities and Choquet's concept.

This booklet can be utilized for a one-semester graduate direction on Convex services and purposes, and likewise as a important reference and resource of suggestion for researchers operating with convexity. the one necessities are a historical past in complicated calculus and linear algebra. every one part ends with routines, whereas every one bankruptcy ends with reviews overlaying supplementary fabric and old info. Many effects are new, and the entire publication displays the authors’ personal adventure, either in educating and research.

About the authors:

Constantin P. Niculescu is a Professor within the division of arithmetic on the collage of Craiova, Romania. Dr. Niculescu directs the Centre for Nonlinear research and Its purposes and in addition the graduate application in utilized arithmetic at Craiova. He acquired his doctorate from the collage of Bucharest in 1974. He released in Banach house thought, Convexity Inequalities and Dynamical structures, and has got a number of prizes either for examine and exposition.

Lars Erik Persson is Professor of arithmetic at Luleå collage of expertise and Uppsala collage, Sweden. he's the director of middle of utilized arithmetic at Luleå, a member of the Swedish nationwide Committee of arithmetic on the Royal Academy of Sciences, and served as President of the Swedish Mathematical Society (1996-1998). He obtained his doctorate from Umeå collage in 1974. Dr. Persson has released on interpolation of operators, Fourier research, functionality thought, inequalities and homogenization concept. He has acquired numerous prizes either for study and teaching.

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**Additional resources for Convex Functions and Their Applications: A Contemporary Approach**

**Example text**

7 If f : I → R is a convex function, then f is Lipschitz on any compact interval [a, b] contained in the interior of I. Proof. 3, f+ (a) ≤ f+ (x) ≤ f (y) − f (x) ≤ f− (y) ≤ f− (b) y−x for all x, y ∈ [a, b] with x < y, hence f |[a,b] veriﬁes the Lipschitz condition with L = max{|f+ (a)|, |f− (b)|}. 8 If fn : I → R (n ∈ N) is a pointwise converging sequence of convex functions, then its limit f is also convex. Moreover, the convergence is uniform on any compact subinterval included in int I, and (fn )n converges to f except possibly at countably many points of I.

13) we get x b (y − a) dµ(y) + a (b − y) dµ(y) = f (a) + f (b) − 2f x a+b , 2 which yields 0≤ 1 b−a (x − a)(b − x) dµ(x) ≤ f (a) + f (b) − 2f I a+b . 15) In the general case we apply the above reasoning to the restriction of f to the interval [a + ε, b − ε] and then pass to the limit as ε → 0. The uniqueness of µ is a consequence of the fact that f = µ in the sense of distribution theory. 6 Integral Representation of Convex Functions 39 for all ϕ ∈ Cc2 (I), which yields f (x)ϕ (x) dx = I G(x, y)ϕ (x) dx dµ(y) = I×I ϕ(y) dµ(y), I due to the symmetry of G and the Fubini–Tonelli theorem.

Riesz exhibited an example of increasing function ϕ with ϕ = 0 almost everywhere. See [103, pp. 278–282]. The corresponding function f in his example is strictly convex though f = 0 almost everywhere. As we shall see in the Comments at the end of this chapter, Riesz’s example is typical from the generic point of view. 1 an important integral representation of all continuous convex functions f : [a, b] → R. For this we need the following Green function associated to the bounded open interval (a, b): G(x, y) = (x − a)(y − b)/(b − a), if a ≤ x ≤ y ≤ b (x − b)(y − a)/(b − a), if a ≤ y ≤ x ≤ b.

### Convex Functions and Their Applications: A Contemporary Approach by Constantin P. Niculescu, Lars-Erik Persson (auth.)

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