are all the norms invariant under permutation of the indices p-norms?
Well, you answered that exact question, but here’s a description of all norms (on a finite dimensional real vector space): a norm determines the set of all vectors of norm less than or equal to 1. This is convex and symmetric under inverting sign (if you wanted complex, you’d have to allow multiplication by complex units). It determines the norm: the norm of a vector is the amount you have to scale the set to envelope the vector. Any set satisfying those conditions determines a norm.
So there are a lot of norms out there. eg, you can take a cylinder in 3-space (one of your examples). You could take a hexagon in the plane. This norm allows the interchange of coordinates, but it has a bigger symmetry group, though still finite. (I guess one could write this as max(|x|,|y|,|x-y|))
are all the norms invariant under permutation of the indices p-norms?
Well, you answered that exact question, but here’s a description of all norms (on a finite dimensional real vector space): a norm determines the set of all vectors of norm less than or equal to 1. This is convex and symmetric under inverting sign (if you wanted complex, you’d have to allow multiplication by complex units). It determines the norm: the norm of a vector is the amount you have to scale the set to envelope the vector. Any set satisfying those conditions determines a norm.
So there are a lot of norms out there. eg, you can take a cylinder in 3-space (one of your examples). You could take a hexagon in the plane. This norm allows the interchange of coordinates, but it has a bigger symmetry group, though still finite. (I guess one could write this as max(|x|,|y|,|x-y|))