Can Principal Moments of Inertia Exceed the Sum of the Others?

knightpraetor
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How do i go about proving that none of the principal moments of inertia can exceed the sum of the other two?

Someone suggested the triangle inequality, but i don't understand how to use it.
 
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Just write out the definitions:

I_x=\int (y^2+z^2)dm
I_y=\int (x^2+z^2)dm
I_z=\int (x^2+y^2)dm

and fiddle around with that.
 

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