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By Krantz S.G.

ISBN-10: 149870638X

ISBN-13: 9781498706384

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Example text

Now set α = y/x, and for fixed t define A(α) = [(1 − t) + tα]4 and B(α) = (1 − t) + tα4 . We note that A(0) = (1 − t)4 < (1 − t) = B(0) . Furthermore, A (α) = 24t3 < 24 = B (α) . Integrating three times, and applying the fundamental theorem of calculus, we see that A(α) < B(α) and f is strictly convex. 46 CHAPTER 2. FUNCTIONS In the converse direction we note that a function with continuous, positive definite Hessian will certainly be strictly convex. We leave the details of that assertion to the reader.

We say that F is strictly (or strongly) convex if the inequality is strict. 18 The function f(x) = ex is strictly convex. To see this, consider the inequality f((1 − t)P + tQ) ≤ (1 − t)f(P ) + tf(Q) . 3 A simple instance of the Minkowski functional is the following. Let K ⊆ RN be convex. For x ∈ RN , define p(x) = inf{r > 0 : x ∈ rK} . Then p is a Minkowski functional for K. 1. 3. 13: The definition of convex function. 14: A convex function that is not strictly convex. 42 CHAPTER 2. FUNCTIONS This becomes or e(1−t)P +tQ ≤ (1 − t)eP + teQ et(Q−P ) ≤ (1 − t) + teQ−P .

The chapter begins with a concept that will be new for many readers: the idea of defining function. This is a key epistemological point in the book— to associate to any reasonable domain a function. The idea is that the function contains all the geometric information about that domain. There are no obvious algebraic operations on domains, but there are many such operations on functions. We can take good advantage of that observation. Using the defining function, we can finally give an analytic definition of convex set.

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Convex analysis by Krantz S.G.

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