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Inner (dot) products

  1. Mar 19, 2008 #1
    1. The problem statement, all variables and given/known data

    For [tex]x,y \in R^n[/tex], their inner ("dot") product is given by

    [tex]<x,y>=\sum_{i=1}^n x_i y_i.[/tex]

    Also, we write


    2. Relevant equations

    Fix p>1. Show that for all [tex]x,y \in R^n[/tex] we have

    [tex]< \|x\|^{p-2}x -\|y\|^{p-2}y, x-y> \geq 0[/tex]

    3. The attempt at a solution

    Expanding the left-hand side, we can write

    [tex]<\|x\|^{p-2}x,x> -<\|y\|^{p-2}y,x>-<\|x\|^{p-2}x,y>+<\|y\|^{p-2}y,y>[/tex]

    which further simplifies to

    [tex]\|x\|^p +\|y\|^p -(\|x\|^{p-2} +\|y\|^{p-2})<x,y>.[/tex]

    Then I'm stuck. How do I show that the above is nonnegative?
    Last edited: Mar 19, 2008
  2. jcsd
  3. Mar 19, 2008 #2
    how about using cauchy-schwartz inequality?
    something like this
    then you get that what you wrote is greater than:
    now if ||x||>=||y|| then ||x||^p-1>=||y||^p-1
    I leave you to check this proposition.
  4. Mar 19, 2008 #3
    I've got it now, many many thanks!
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