A (somewhat) common problem in geometry and optimization deals with how to most efficiently pack $n$ rigid disks inside a given container of some fixed size and shape (e.g. a circular container, a rectangular container, etc). I'm curious whether any work has been done on the question of circle packing in a container with an elastic boundary, the shape of which is allowed to vary.
It seems to me that, if we allow a collection of circles to be enclosed in an elastic container, the most efficient packing should be the packing that minimized the stretching of the elastic - i.e. its perimeter. The question, then, can be stated:
Given a collection of $n$ unit disks, find a placement of the disks and a closed piecewise smooth curve such that
- None of the disks overlap
- Every disk is enclosed by the curve
- The length of the curve is minimized
Obviously, this is a very difficult problem, so I don't expect any solutions below. However, I'd be grateful for any references on existing work on this question (if any exist).