Richard Schroeppel asked:
Puzzle: Choose some specific ellipse, say a=2 b=1. Find any packing that has a higher packing density than circles in the plane (the simple hex packing).
And I generalized the question to:
Suppose we're allowed to use more than one shape of ellipse, as long as all of them have the same area. Can we get a higher density than the hex packing of circles?
I asked Sherman Stein, who asked Don Chakerian, who found a reference that gives negative answers to both questions:
I found in Fejes-Toth's Lagerungen in der Ebene, auf der Kugel, und im Raum, a proof that the density of the closest packing of plane convex sets all congruent to a given set K is at most area(K)/area(H), where H is a hexagon of least area circumscribed about K. Then on p.88 he remarks that his proof extends to equiareal affine images of a given convex set, from which one deduces that ellipses of equal area cannot be packed denser than congruent circular disks.
I don't have access to a good math library, so I won't be able to look for the book. Dean Hickerson dean@math.ucdavis.edu