@Daniel: How do we connect points when $P$ has to be $n$-times differentiable?

Thanks. Can't we extend this for multiple $q$'s, by having $F(\gamma)= length(\gamma) + \sum inf_{x} |p-q_i|^2$ ? And under what conditions can we differentiate through the infimum?

@ fedja: Won't the huge "almost circles" [ whatever that might be] lead to a big increase in length?

Right, from a programmer's perspective it's a dynamic programming problem. However, I'm more interested in the <i>formulation </i> of the problem, and trying to extend this to functionals that may not be possibly additive.

Well, I always thought that tagging in MathOverflow (unlike StackOverflow) was a matter of "best approximation".