Suppose I have two polynomial functions $f(x)$ and $z(x)$ defined over some closed interval $[a,b]$ $\in$ $\mathbb{R}$. The functions are subject only to the restriction that $f(x) > z(x)$ $\forall$ $x$.
I am trying to find a function $g(x)$ that meets the following conditions for any closed interval $[a, b]$ on $x$:
Condition 1: $f(x) > g(x) > z(x)$
This ensures that $g(x)$ never "touches" either function, being bounded above by $f(x)$ and bounded below by $z(x)$
Condition 2: Does so with the "shortest" line possible subject to Condition 1.
I'm not sure how to precisely define "shortest" in this context, but if someone walked along the line, the total distance travelled would be the smallest. I think about this in terms of the typical GAM/Penalized Spline penalty on "squiggliness": minimizing something like $(g(x+h) - g(x))^2$ $\forall$ $x \in [a,b]$.
How would I go about finding $g(x)$?
Substantive Motivation/Application
A substantive motivation for this question occurs in racing athletics: the athlete must travel from the starting line to the finishing subject to staying on the course. The course can be arbitrarily wide or narrow at different points. Assuming the course is homogenous, the athlete seeks to find the shortest route. While this seems like a Traveling Salesman problem, I'm not sure if the upper and lower bounds make this more tractable or provide a different solution.