But the cube isn't pegged like that at all. The cube is the set of points where any element of the coordinate vector has a magnitude less than a specified value

But the cube isn't pegged like that at all. The cube is the set of points where any element of the coordinate vector has a magnitude less than a specified value, whereas the sphere is the set of points where the magnitude of the vector is less than a specified value. Hence the cube gets 'larger' as you make space itself larger by adding dimensions, but the sphere doesn't.