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Identifing a vector from all Lp norms ?

  1. Sep 25, 2007 #1
    identifing a vector from all Lp norms....?

    Hey fellows,

    If I have all the [tex]L_p[/tex] norms for all p in natural numbers or positive reals of a function from real to real, can I identify the function up to some equivalence relation (such as permutation of the ) assuming the function has a finite [tex]L_0[/tex] norm?

    The discrete version of this question would be: can I identify a vector [tex]x \in \mathbb{R}^N[/tex] given all the [tex]l_p[/tex] norms defined as follows:
    [tex]
    l_p(x) = \left(\sum_i^N |x_i|^p \right)^{\frac{1}{p}}
    [/tex]
    It is apparent from the equation that the permutation of a vector will result in the same norm.

    Any ideas?
     
  2. jcsd
  3. Sep 25, 2007 #2

    EnumaElish

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    It would seem that you can at most hope to identify the absolute values of the vector elements.
     
  4. Sep 25, 2007 #3
    For the discrete case at least you can try visualising surfaces of constant norm. Constant L_1 norm gives hyperplanes. Constant L_2 norm gives hyperspheres. Constant L_inf gives a box. So increasing p gradually pushes the surface outwards. Notice that the surfaces all have a line of symmetry going through x_0=x_1=x_2=... I'm not entirely sure, but my intuition says that yes, you can work out the absolute value of the components up to a permutation of them.
     
  5. Sep 26, 2007 #4
    Yes, my intuition also says so. But I cannot prove it!!
    The geometric picture helps in low dimensions, but I am having great trouble imagining N-1 hyperstructures intersecting with each other....

    If you have any reference to the literature, it would be very helpful.
     
  6. Nov 22, 2007 #5
    Counter example

    I found a counter example in the continuous case.
    A exponential function and a piecewise-patched Laplacian distribution function has the same L-alpha norm, and you can make them have the same range [0, [tex]\infty[/tex]].
     
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