By Jonathan M. Borwein

Like differentiability, convexity is a normal and robust estate of capabilities that performs an important function in lots of components of arithmetic, either natural and utilized. It ties jointly notions from topology, algebra, geometry and research, and is a crucial device in optimization, mathematical programming and video game idea. This ebook, that's the made of a collaboration of over 15 years, is exclusive in that it makes a speciality of convex capabilities themselves, instead of on convex research. The authors discover some of the periods and their features and purposes, treating convex features in either Euclidean and Banach areas. The ebook can both be learn sequentially for a graduate path, or dipped into by way of researchers and practitioners. each one bankruptcy includes a number of particular examples, and over six hundred workouts are integrated, ranging in trouble from early graduate to investigate point.

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Extra resources for Convex Functions: Constructions, Characterizations and Counterexamples (Encyclopedia of Mathematics and its Applications)

Example text

Let f (x, y) := g(x, y) := smooth on xy3 x2 +y4 R2 \ xy4 x2 +y8 if (x, y) = (0, 0) and let f (0, 0) := 0; let if (x, y) = (0, 0) and g(0, 0) := 0. Then, both functions are C ∞ {(0, 0)}. 5 Functions 2 4 and 2 8 with a Gâteaux derivative at the origin that x +y x +y is not Fréchet. continuous there, while g is Gâteaux differentiable and continuous at (0, 0), but not Fréchet differentiable there. Proof. The reader can check that the Gâteaux derivatives of both functions at (0, 0) are the zero functional.

3. Suppose f : E → [−∞, +∞] is convex and that some point x0 ∈ core dom f satisfies f (x0 ) > −∞. Then f never takes the value −∞. Proof. Suppose f (x0 − h) = −∞ for some h ∈ X . The convexity of the epigraph of f then implies f (x0 + th) = ∞ for all t > 0. This contradicts that x0 ∈ core dom f . Let E and Y be Euclidean spaces and A : E → Y a linear mapping. The adjoint of A, denoted by A∗ , is the linear mapping from Y to E defined by A∗ y, x = y, Ax for all x ∈ E. 4 (Fenchel duality theorem).

Proof. 1. 1. 9. 2. Suppose f : R → (0, ∞), and consider the following three properties. (a) 1/f is concave. (b) f is log-convex, that is, log ◦f is convex. (c) f is convex. Show that (a) ⇒ (b) ⇒ (c), but that none of these implications reverse. Hint. 2]; to see the implications don’t reverse, consider g(t) = et and h(t) = t respectively. 3. Prove that the Riemann zeta function, ζ (s) := n=1 s is log-convex on n (1, ∞). 4. Suppose h : I → (0, ∞) is a differentiable function. Prove the following assertions.

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