P-norms for p<1

  • #1
Office_Shredder
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I'm interested in thing that are norms except for the fact that they satisfy the reverse triangle inequality [tex] ||x+y|| \geq ||x|| + ||y||[/tex]. The obvious example is taking p-norms for 0<p<1. Does anyone know of others or if there's any theory developed on this topic?
 

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  • #2
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Are you requiring the norm to be positive definite? If so, I'm pretty sure that your space can have only one point.


EDIT: the p norms don't satisfy the reverse triangle inequality: take x = -y.
 
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  • #3
Office_Shredder
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Oops you're right. I was reading on wikipedia and misinterpreted something.

I should have just stuck with what I originally wanted, which is a "norm" whose unit ball is as concave as possible, rather than convex. Obviously you can't have a unit ball that never contains a line between two points (since it's centrally symmetric)
 

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