Identifing a vector from all Lp norms ?

[memming]

identifing a vector from all Lp norms...?

Hey fellows,

If I have all the $$L_p$$ 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 $$L_0$$ norm?

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

Any ideas?

 

[EnumaElish]

It would seem that you can at most hope to identify the absolute values of the vector elements.

 

[genneth]

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.

 

[memming]

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.

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.

 

[memming]

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, $$\infty$$].