Here is a problem that I have been stuck with:
Given a simple closed curve $\alpha _1$, the curve parallel to it $\alpha _2$ and a distance $ε$ between any two points with common normal (green on the picture).
Find the shortest simple closed curve which lies between the parallel curves (red on the picture).
I would be grateful for any advice. There are papers around with a discrete algorithm solution, but I'm interested only in an analytic solution.