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Homework Help: Convex Function on Many variables on an interval

  1. Nov 3, 2012 #1
    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 [itex]\geq[/itex] 0
    and m [itex]\geq[/itex] 1.

    2. Relevant equations
    By definition f is convex iff

    [itex]\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)[/itex]

    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. jcsd
  3. Nov 3, 2012 #2

    Ray Vickson

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    Science Advisor
    Homework Helper

    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.

  4. Nov 3, 2012 #3
    Yes you are right in this, sorry I did not stipulate this, x1= a and x2=ma=mx1
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