By Jonathan M. Borwein

Like differentiability, convexity is a average and strong estate of services that performs an important position in lots of parts of arithmetic, either natural and utilized. It ties jointly notions from topology, algebra, geometry and research, and is a crucial software in optimization, mathematical programming and video game thought. This booklet, that is the fabricated from a collaboration of over 15 years, is exclusive in that it specializes in convex services themselves, instead of on convex research. The authors discover a few of the periods and their features and functions, treating convex services in either Euclidean and Banach areas. The ebook can both be learn sequentially for a graduate direction, or dipped into via researchers and practitioners. each one bankruptcy incorporates a number of particular examples, and over six hundred workouts are integrated, ranging in hassle from early graduate to analyze point.

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**Additional info for Convex Functions: Constructions, Characterizations and Counterexamples**

**Example text**

12) 0 for Re(x), Re(y) > 0. As is often established using polar coordinates and double integrals (x) ( y) β(x, y) = . 13) for real x, y. 24 are easy verify. 3). Thus f = as required. (b) Show that the volume of the ball in the · p -norm, Vn (p) is Vn (p) = 2 n (1 + p1 )n (1 + pn ) . 14) as was first determined by Dirichlet. When p = 2, this gives Vn = 2n ( 32 )n = (1 + n2 ) ( 12 )n , (1 + n2 ) which is more concise than that usually recorded in texts. Maple code derives this formula as an iterated integral for arbitrary p and fixed n.

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)}.

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. (a) 1/h is concave ⇔ h( y) + h ( y)(y − x) ≥ h( y) 2 h(x), for all x, y in I .