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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).

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My advice would be to start with a class of bounding curves, the curves of constant width (CCW). Read : https://en.wikipedia.org/wiki/Curve_of_constant_width.

From Barbier's theorem, perimeter is $(\pi d)$ for all curves in the family with diameter $d$. Try to solve the simpler problem to pack $n$ disks inside such a curve, for example the Reuleaux triangle or the Stanley Rabinowitz's curve (end of wikipedia paper). Then try the the star construction of CCW. Inversive geometry could be your friend.

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