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Proving a convex function on an open convex set satisfies some inequalities

  1. Mar 7, 2010 #1

    michael.wes

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    Gold Member

    1. The problem statement, all variables and given/known data
    Let [tex]f:\mathcal{O}\subset\mathbb{R}^n\rightarrow\mathbb{R}, \mathcal{O}[/tex] is an open convex set. Assume that [tex]D^2f(x)[/tex] is positive semi-definite [tex]\forall x\in\mathcal{O}[/tex]. Such [tex]f[/tex] are said to be convex functions.


    2. Relevant equations
    Prove that [tex]f((1-t)a+tb)\leq (1-t)f(a)+tf(b),a,b\in\mathcal{O},0 \leq t \leq 1[/tex] and interpret the result geometrically. (The interpretation is easy, it's the proof of the inequality I'm stuck with)


    3. The attempt at a solution

    In an earlier part of the question I proved that we have


    [tex]f(x)\geq f(a) + \grad f(a)\cdot (x-a) \forall x,a\in\mathcal{O}[/tex]

    I have tried to use the mean value theorem in R^n in an attempt to link this with that result and the gradient, but it doesn't help since you lose information in using the mean value theorem, and this isn't an existence result, so it makes me think that there is a direct approach. This is typically given as the definition of a convex function on the web, and not a theorem, so I couldn't find help elsewhere.

    Any help appreciated!

    Edit: I'm still completely stuck.
     
    Last edited: Mar 7, 2010
  2. jcsd
  3. Mar 7, 2010 #2

    michael.wes

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    Gold Member

    Any help before tomorrow would be appreciated :)
     
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