What is the difference between norm of second derivative and curvature?
Good question. In Schoenberg's case (if I remember correctly), he approximates a curve by its height over a given tangent line, because he only wants to consider small displacements. What's the difference between its second derivative of a height function and the curvature of its graph? Quite a bit, actually; take a look at the Wikipedia article on curvature for some explicit calculations.