414/514 – Convexity

This is the last homework set of the term. It is optional. If you decide to turn it in, it is due Wednesday, December 14 at noon.

1.

We say that a function f:{\mathbb R}\to{\mathbb R} satisfies Cauchy’s functional equation iff

f(x+y)=f(x)+f(y)

for all x,y. Show that f(0)=0 and that there is a constant c such that f(x)=cx for all rational x. One way to organize the argument is as follows: First show that there is such a c if x is a natural number, by induction. Then show that the same c works if x is an integer (positive, negative, or zero), then that it works if x is the multiplicative inverse of an integer, and finally for arbitrary rational numbers.

Show that if f is continuous, then the same holds for all reals x. Show that it suffices to assume that f is continuous at 0 to conclude that it is continuous everywhere. What if we assume that f is continuous at a point x_0 (but x_0 could be different from 0)? Can we conclude that f is continuous everywhere in this case as well? (On the other hand, using the axiom of choice one can produce examples of discontinuous functions that satisfy Cauchy’s functional equation.)

2.

Suppose that f is a function defined on an interval, and that for any x and y in its domain, and any t\in[0,1], we have that

tf(x)+(1-t)f(y)\ge f(tx+(1-t)y).

If this is the case, we say that f is convex. Draw a diagram explaining the definition. Convexity is a very useful notion in analysis, especially when combined with additional conditions, such as harmonicity.

Show that if f is continuous and satisfies the condition above for t=1/2, then it is convex.

Show that if f is convex, then it is continuous, except perhaps at the end points of the interval. In any case, if f is convex, continuous, and it is defined at an end point a of the interval, show that the limit of f(x) as x\to a exists (and is finite).

Suppose that f is convex and that a\le b<c\le d are points in the interval. Show that

\displaystyle \frac{f(c)-f(a)}{c-a}\le \frac{f(d)-f(b)}{d-b},

and explain what this says about the graph of f.

Show that if f is convex and differentiable, then f' is increasing.

Show that if f is convex then whether f is differentiable or not, its left and right derivatives exist at every point in the interior of the interval, and are increasing. Here, the left derivative of f at x is

\displaystyle \lim_{h\to 0^-} \frac{f(x+h)-f(x)}{h}

and similarly for the right derivative (replace 0^- with 0^+).

Show that if f is twice differentiable on an open interval, then f is convex iff f''(x)\ge0 for all x in the interval.

One of the main reasons why convexity is a useful property is the following theorem of Jensen. Please give a proof: Suppose that f is convex and x_1,\dots,x_n are nonnegative numbers that add up to 1. Then for any t_1,\dots,t_n in the domain of f,

x_1 f(t_1)+\dots+x_n f(t_n)\ge f(x_1 t_1+\dots+x_n t_n).

As an application, prove the inequality between the arithmetic and the geometric means: If a_1,\dots,a_n are nonnegative numbers, then

\displaystyle \frac{\sum_{i=1}^n a_i}n\ge\root n\of{\prod_{i=1}^n a_i}.

(It may be useful to prove first that \exp is a convex function.)

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