Convexity of functions

Question

I'm trying to read up on convex/concave functions (is that the same as concave up, concave down?)

• If I were asked to prove a convexity of a function, what are the general steps to follow? (So far I know by standard definition, to form the conclusion based on the second derivative)

• In case I have a composite function: "-ln$g(x)$" , how do I prove that this is neither concave/convex ?

Answer 1

as you already said, looking at the second derivative is one of the main technices for prooving that a function is convex. As far as i know it suffices for continues functions to show, that $\forall a,b :\frac{f(a)+f(b)}{2} \geq f(\frac{a+b}{2})$ holds. about the function $-ln(g(x))$ (i guess that is what you wanted to write): if you want to prove that smth is not convex just look out if you can find a counterexample to the definiten, e.g. three values $a,b,c$ such that $a+b=2c$ and $f(a)+f(b)<2f(c)$

if you would tell us what g is, we could help you maybe further

Answer 2

One of the goals of the book Convex Optimization by Boyd and Vandenberghe (free online) is to teach people to recognize convex functions. Aside from directly checking the definition or checking the second derivative, there are a bunch of ways of combining functions to create new convex functions. For example, a supremum of convex functions is convex. A conic combination of convex functions is convex. Any norm is convex. If $f$ is convex and $T$ is affine, then $g(x) = f(T(x))$ is convex. There are various chain rules that can guarantee convexity. There is a partial minimization rule that states that if $f(x,y)$ is a convex function on $X \times Y$ (where $X$ and $Y$ are convex sets), then $g(y) = \inf_{x \in X} f(x,y)$ is convex under certain assumptions.