# Homework Help: Convex Function on Many variables on an interval

1. Nov 3, 2012

### curiousguy23

1. The problem statement, all variables and given/known data
Show that f(x) = x1x2 is a convex function on [a,ma]T where a $\geq$ 0
and m $\geq$ 1.

2. Relevant equations
By definition f is convex iff

$\forall x,y\in \Re \quad \wedge \quad \forall \lambda :\quad 0\le \lambda \le 1\quad \Rightarrow \quad f\left( \lambda x+(1-\lambda )y \right) \le \lambda f\left( x \right) +(1-\lambda )f\left( y \right)$

3. The attempt at a solution

I am not really sure how to go about this. Firstly I was thinking on how to apply the above relation to multiple variables. I would assume that it applies in general and in this case x and y would be a vector of variables, but I still don't know how the proof follows. Also I am not sure is how the interval [a,ma]T plays into the equation.

Last edited: Nov 3, 2012
2. Nov 3, 2012

### Ray Vickson

What is meant by [a,ma]T? I doubt that [a,ma] is a 1-dimensional interval, because what in the world would possibly be meant by the transpose of an interval? My guess would be that [a,ma] means a point in R2 of the form x1 = a, x2 = ma. However, that is just a guess.

RGV

3. Nov 3, 2012

### curiousguy23

Yes you are right in this, sorry I did not stipulate this, x1= a and x2=ma=mx1