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P-norms for p<1

  1. Jun 10, 2010 #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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  3. Jun 10, 2010 #2
    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.
     
    Last edited: Jun 10, 2010
  4. Jun 10, 2010 #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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