# Minimize function curve 'length'

Given a set of points $(x_i, y_i)$, how can I find the serie of $C^\infty$-functions for which the sum passes through all points and for which the length of the resulting curve is minimal; i.e. if you see the function graphical representation as a path, I want the shortest path.
It feels like I am trying to find the solution which minimizes local convexities, but I cannot prove it.

• Are you sure you want to 'pass through all points'? If the (x,y) pairs are at all noisy this is likely to be a bad idea, if not impossible. E.g. the Wiener process is not differentiable so there isn't going to be a smooth function passing through all the points. Do you perhaps mean 'passes close to all points'? – conjectures Jun 28 '18 at 13:20
• In my case I mean passes through all points indeed. My points are not random and the overall function is fairly monotonous. Would it help if I gave you a numerical example ? – Djiggy Jun 28 '18 at 13:34
• If you are looking for a single function, Euclid taught us the solution (regardless of differentiability) must consist of line segments. Since you assume these will form the graph of a function, that function must be piecewise linear. Since (1) a piecewise linear (but nonlinear) graph is not $C^\infty$ but (2) can be approximated arbitrarily closely with a $C^\infty$ graph, there is no shortest solution when your points are not collinear, but there are solutions that are as close to the shortest as you like. That's why you cannot prove much. But what would a series of functions mean? – whuber Jun 28 '18 at 15:28
• ha, maybe my wording is not correct, for me an example of a "serie of functions" would be a Fourier-serie or Chebyshev polynomials. – Djiggy Jun 28 '18 at 15:34