Convex, continuous, Jensen’s
On the principle that things I’ve forgotten at least thrice are worth remembering…
A convex function (need a mnemonic?) is such that
for any λ in [0,1].
Midpoint convex (almost) implies convex
A natural question is whether we need all λ in [0,1]; what about just ½? Functions which satisfy
are called midpoint convex.
It turns out that when things are “nice” as in most frequently encountered cases, a midpoint convex function is convex. (A counterexample is this crazy non-measurable nowhere-continuous bounded-in-no-interval function: since R is a vectorspace over Q, there is a basis (that includes π, say), and any real number x has a unique representation as a finite rational sum of basis elements. Define f(x) to be the coefficient of π in the representation of x. This function is in fact linear over Q.)
For midpoint convex to mean convex, any of the following conditions will do:
- f is continuous [Proof: prove the inequality for λ with denominators that are powers of 2, then use denseness in R.]
- f is continuous at some point (any one point will do) [Proof: see next section]
- f is (Lebesgue) measurable
- f is bounded on some measurable set with positive measure
Convex functions are continuous
In fact, a stronger theorem is true: If a midpoint convex function is discontinuous at a single point, then it is unbounded on every interval and (hence) discontinuous everywhere.
Proof: Suppose wlog that f is discontinuous at 0, that f(0)=0, and that a sequence has for some positive m. Prove that , so f is unbounded near 0. Then use this to show unbounded everywhere.
Note that convex functions are bounded on every interval, so by above theorem they are continuous.
Convex functions are differentiable almost everywhere
In fact a convex function has left and right derivatives at all points, these are monotonically increasing, and there are only countably many points where they don’t coincide.
All the above is from pages 10 to 16 of this book I had found through googling; if anyone’s reading this, please tell me if you know better proofs.
A proof of Jensen’s inequality
Here, Jensen’s inequality is the statement that for a convex function ,
An equivalent definition of a convex function is that at any point we can find a line through the point that lies entirely below the function. (Why?) In other words, for any point , there is a “slope” m such that
for all .
With this definition, Jensen’s inequality has an absurdly short proof: take to be the mean , then take expectation.