2017/06/16 by Böröczky, K., Heppes, A., Makai, E.
#52A43 #52A45 #FOS: Mathematics #Metric Geometry (math.MG) #Primary: 52C17 #Secondary: 52C07
paper · doi:10.48550/arxiv.1706.05282
Let L ⊂ \Bbb R3 be the union of unit balls, whose centres lie on the z-axis, and are equidistant with distance 2d ∈ [2, 2√(2)]. Then a packing of unit balls in \Bbb R3 consisting of translates of L has a density at most π/(3d√(3-d2)), with equality for a certain lattice packing of unit balls. Let L ⊂ \Bbb R4 be the union of unit balls, whose centres lie on the x3x4 coordinate plane, and form either a square lattice or a regular triangular lattice, of edge length 2. Then a packing of unit balls in \Bbb R4 consisting of translates of L has a density at most π2/16, with equality for the densest lattice packing of unit balls in \Bbb R4. This is the first class of non-lattice packings of unit balls in \Bbb R4, for which this conjectured upper bound for the packing density of balls is proved. Our main tool for the proof is a theorem on (r,R)-systems in \Bbb R2. If R/r ≤ 2 √(2), then the Delone triangulation associated to this (r,R)-system has the following property. The average area of a Delone triangle is at least min \ V0, 2r2 \ , where V0 is the infimum of the areas of the non-obtuse Delone triangles. This general theorem has applications also in other problems about packings: namely for 2r2 ≥ V0 it is sufficient to deal only with the non-obtuse Delone triangles, which is in general a much easier task. Still we give a proof of an unpublished theorem of L. Fejes Tóth and E (=J.) Székely: for the 2-dimensional analogue of our question about equidistant strings of unit balls, we determine the densest packing of translates of an equidistant string of unit circles with distance 2d, for the first non-trivial interval 2d ∈ (2√(3),4).