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Convex Functions: Constructions, Characterizations and by Jonathan M. Borwein

By Jonathan M. Borwein

Like differentiability, convexity is a typical and robust estate of capabilities that performs an important function in lots of parts 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 thought. This ebook, that's the manufactured from 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 a number of the periods and their features and functions, treating convex features in either Euclidean and Banach areas. The publication can both be learn sequentially for a graduate path, or dipped into by way of researchers and practitioners. each one bankruptcy incorporates a number of particular examples, and over six hundred routines are incorporated, ranging in trouble from early graduate to analyze point.

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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. (a) 1/h is concave ⇔ h(y) + h ( y)(y − x) ≥ h( y) 2 h(x), for all x, y in I . 2 (b) 1/h is affine ⇔ h(y) + h ( y)( y − x) = h( y) h(x), for all x, y in I . (c) 1/h is concave ⇒ h is log-convex ⇒ h is convex. (d) If h is twice differentiable, then: 1/h is concave ⇔ hh ≥ 2(h )2 . 5. Let U ⊂ E be an open convex set, and let f : U → R be a convex function.

If the function f : →R Proof. 10 , it suffices to show that f is bounded above on . For this, let x ∈ . Then n f (x) = f n xi ei + 1 − xi f (ei ) + 1 − xi 0 ≤ i=1 xi f (0) i=1 ≤ max{ f (e1 ), f (e2 ), . . , f (en ), f (0)}, where {e1 , e2 , . . , en } represents the standard basis of Rn . 12. Let f : E → (−∞, +∞] be a convex function. Then f is continuous (in fact locally Lipschitz) on the interior of its domain. Proof. For any point x ∈ int dom f we can choose a neighborhood of x ∈ dom f that is a scaled and translated copy of the simplex.

Show that g (t) = φt , h except for possibly countably many t ∈ [0, 1]. Hint. First, g is differentiable except at possibly countably many t ∈ [0, 1], and at points of differentiability ∇g(t) = {∂g(t)}. For each t ∈ [0, 1], observe that φt , sh ≤ f (x + (s + t)h) − f (x + th) = g(t + s) − g(t). Hence φt , h ∈ ∂g(t). 6 (A compact maximum formula [95]). Let T be a compact Hausdorff space and let f : E × T → R be closed and convex for in x ∈ E and continuous in t ∈ T . Consider the convex continuous function fT (x) := max ft (x) where we write ft (x) := f (x, t), t∈T and let T (x) := {t ∈ T : fT (x) = ft (x)}.

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