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Math Help - Problem with convex curve and Jensen inequality

  1. #1
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    Problem with convex curve and Jensen inequality

    given a convex curve function f(x),

    Jensen inequality states:
    E[ f(x) ] \geq f( E[x] )

    Preliminary:

    Let l(x) be linear function below f(x). Since a convex function is the maximum of all linear functions that lie below it:

    f(x) \geq max\{l(x)\}
    >>> this equation I understand.

    On the other hand, because f(x) is convex, there is always a value x for which f(x) = l(x), where l(x) is the tangent (or support line) to f(x).

    So,
    f(x) \leq max\{l(x)\}
    >>> I don't understand at all.

    So f(x) = max{l(x)}.
    This is a relationship we are going to use to prove Jensen's inequality.

    Can anyone explain the problem I have with the 2nd inequality first before I post the rest of the solution (which I can follow.) ?

    Thanks.
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  2. #2
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    Quote Originally Posted by chopet View Post
    given a convex curve function f(x),

    ...

    On the other hand, because f(x) is convex, there is always a value x for which f(x) = l(x), where l(x) is the tangent (or support line) to f(x).

    So,
    f(x) \leq max\{l(x)\}
    >>> I don't understand at all.

    So f(x) = max{l(x)}.
    ...
    Hello,

    I've attached a sketch to demonstrate what this inequality means
    Attached Thumbnails Attached Thumbnails Problem with convex curve and Jensen inequality-jensen_unglg2.gif  
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  3. #3
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    Hi, thanks for your sketch. I really appreciate it.
    But the text says convex graph, and yours show a concave graph.

    Anyway, I've reproduced the actual text (from google book) below.
    The offending equation is circled in red.

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