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**1. Homework Statement**

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

[tex]<x,x>=\|x\|^2.[/tex]

**2. Homework 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?

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