The recent discussion re packing of various equiareal ellipses reminds me of a question discussed in this venue some years ago -- but I don't recall the upshot: Which bounded planar convex shape W, packed as well as possible, has the lowest density? (I'm not sure that there's always a packing that realizes the sup over all packings of a given planar shape, so just in case there isn't, let's say we're looking for the shape W whose sup packing density is least over all shapes. And it seems quite possible that there could be many W's tied for this dubious honor. So the real question is, What are all of these? (We want a shape whose sup-best packing density is least among all bdd convex planar shapes is least. Since the space of bdd convex shapes is closed, it seems there will always be an actual such shape that realizes the inf of such sup-best densities.) ------------------------------------------------------------------ As a start, since the sups of a shape K and any affine transformation T applied to it are the same, let's only consider shapes which cannot be made "rounder" by any affine tsfmn. Among many ways to define this, perhaps an affine tsfmn T that makes T(K)'s 2D correlation matrix equal to the identity (when T(K) is viewed as a uniform probability density in R^2) would be the way to go. (What is the word for this matrix in physics?) Now, there may be a natural 2nd-order measure of roundness, and I'd suspect that the worst packer among these "uncorrelated" shapes would be among those whose "2nd-order roundness" is minimum. One obvious 2nd-order measure of roundness is the standard deviation of the radius, as measured from the center of gravity as a function of angle. (To normalize such measures, we should probably assume the average radius -- as a function of angle from the c.o.g. -- is 1.) (On the other hand, such an uncorrelated shape with highest s.d. may well be an equilateral triangle -- which doesn't pack that badly at all.) --Dan