Scott Aaronson proved that for any p other than 1 or 2, the only norm-preserving linear transformations are the permutations of the components.
This seems to be true, but with the small note that you should add multipication of the coordinates by −1 [by any number from unit circle if the space is taken over complex numbers] and their compositions with permutations to the allowed isomorphisms. Never heard about this though, interesting.
However this does not generalize to all the norms. As Douglas noted below one can imagine norm simply as a central-symmetric convex body. And there are plenty of those. Now if we can fix a finite subgroup of space rotations and symmetries that strictly contains all the coordinate permutations and central-symmetry then we are done, since one can simply take convex hull of the orbit of some point as your desired norm.
Symmetries and rotations of regular 100-gon on the plane would work for example.
If you choose the 1-norm, then the sum of the absolute values of the components are preserved, and the norm preserving transformations correspond to the stochastic matrices.
Hmm, something fishy is going with signs in the whole argument and here I am completely lost. What if I take 2x2 matrix with all entries equal to 1⁄2 and a vector (1/2, −1/2)? Probably the full formulation by Scott would help. Does anybody have a link?
Nice paper. Signs are treated accurately there of course.
However call to “formal functions” in the end of the proof seems wacky at best. Formalizing it looks harder to me than the initial statement. At this point it should be easier to just look at the smoothness degrees of the norm on x_i = 0 hyperplanes.
If anybody knows what was meant, however, please clarify.
This seems to be true, but with the small note that you should add multipication of the coordinates by −1 [by any number from unit circle if the space is taken over complex numbers] and their compositions with permutations to the allowed isomorphisms. Never heard about this though, interesting.
However this does not generalize to all the norms. As Douglas noted below one can imagine norm simply as a central-symmetric convex body. And there are plenty of those. Now if we can fix a finite subgroup of space rotations and symmetries that strictly contains all the coordinate permutations and central-symmetry then we are done, since one can simply take convex hull of the orbit of some point as your desired norm. Symmetries and rotations of regular 100-gon on the plane would work for example.
Hmm, something fishy is going with signs in the whole argument and here I am completely lost. What if I take 2x2 matrix with all entries equal to 1⁄2 and a vector (1/2, −1/2)? Probably the full formulation by Scott would help. Does anybody have a link?
This.
Thank you.
Nice paper. Signs are treated accurately there of course. However call to “formal functions” in the end of the proof seems wacky at best. Formalizing it looks harder to me than the initial statement. At this point it should be easier to just look at the smoothness degrees of the norm on x_i = 0 hyperplanes.
If anybody knows what was meant, however, please clarify.