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For any vector norm f(x), if f([0,1,2]) = f([2,1,0]), then f([0,1,2]) = 0.5*(f([0,1,2]) + f([2,1,0]) >= 0.5(f([2,2,2]) = f([1,1,1]), so, in the case of symmetry with respect to coordinates, for vector norms [0,1,2] is at least as big as [1,1,1].
People sometimes talk about a "0-norm", which is the number of non-zero coordinates of the vector. This isn't a genuine vector norm, because it doesn't satisfy f(ax) = a f(x), but it does make f([1,1,1]) = 3 > 2 = f([0,1,2]).
