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What is the history of the result that in a packing of the plane by unit disks, no Delaunay cell can be occupied by disk-sectors whose total measure exceeds $\pi/\sqrt{12}$ times the area of the cell?

I saw this in a two-page article that gave a nice simple proof, but (a) I can no longer locate the article, and (b) it was unclear to me when I read it whether the two authors (Taiwanese mathematicians, I think) were claiming that they were the first to prove the claim, or that their proof was significantly different from earlier proofs.

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  • $\begingroup$ I think Axel Thue proved this in order to establish that the hexagonal lattice is the optimal packing of unit discs in the plane. $\endgroup$ – Adam P. Goucher Feb 26 '18 at 1:48
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    $\begingroup$ Thue's proof is described in some later literature as having been incomplete; I've read that Laszlo Fejes-Toth gave the first rigorous, complete solution to the sphere packing problem in two dimensions. $\endgroup$ – James Propp Feb 26 '18 at 15:44
  • $\begingroup$ I think you will find the reference in: Rogers, C. A., Packing and covering, Cambridge Tracts in Mathematics and Mathematical Physics, No. 54 Cambridge University Press, New York 1964. (MR0172183). The so-called "simplex bound" is proved there for every dimension $n\ge2$, but it is sharp in dimension $2$ only. $\endgroup$ – Wlodek Kuperberg Feb 27 '18 at 3:17
  • $\begingroup$ Also, it is known that in dimension $2$, for any packing with congruent circles, no Voronoi region is of area smaller than the area of the regular hexagon circumscribed about the circle. This was proved by György Hajós, if I remember correctly, but I do not have a reference to that. $\endgroup$ – Wlodek Kuperberg Feb 27 '18 at 3:34
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    $\begingroup$ "I saw this in a two-page article that gave a nice simple proof, but..." – I think you mean this article, Jim: arxiv.org/abs/1009.4322 . $\endgroup$ – Wlodek Kuperberg Feb 27 '18 at 4:01

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