On Mar 28, 2014, at 2:58 PM, Dan Asimov <dasimov@earthlink.net> wrote: I also seem to recall that there were some adjustments to the octagon that give a convex shape with an even lower packing density than the octagon. I don't know if the inf over all convex shapes S of the sup over all density-defined packings' densities is realizable by a specific shape (though I would expect so) or by a specific packing of that shape (no idea). Does anyone out there know more about this? Reinhardt conjecture: http://arxiv.org/abs/1103.4518 -Veit "In 1934, Reinhardt asked for the centrally symmetric convex domain in the plane whose best lattice packing has the lowest density." Doesn't this disregard, e.g., oddgons and Reuleaux triangles? --rwg