We study translative arrangements of centrally symmetric convex domains in the plane (resp., of congruent balls in the Euclidean 3-space) that neither pack nor cover. We define their soft density depending on a soft parameter and prove that the largest soft density for soft translative packings of a centrally symmetric convex domain with 3-fold rotational symmetry and given soft parameter is obtained for a proper soft lattice packing. Furthermore, we show that among the soft lattice packings of congruent soft balls with given soft parameter the soft density is locally maximal for the corresponding face centered cubic (FCC) lattice.
F. Aurenhammer, R. Klein and D.-T. Lee. Voronoi diagrams and Delaunay triangulations. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013.
K. Bezdek and Z. Lángi. Density bounds for outer parallel domains of unit ball packings. Proc. Steklov Inst. Math., 288(1):209–225, 2015.
K. Böröczky. Closest packing and loosest covering of the space with balls, Studia Math. Sci. Hungar. 21 (1986), 79–89.
K. Böröczky and L. Szabó. 12-neighbour packings of unit balls in 𝔼3. Acta Math. Hungar., 146(2):421–448, 2015.
B. Csikós. On the volume of the union of balls. Discrete Comput. Geom., 20:449–461, 1998.
H. Edelsbrunner and M. Iglesias-Ham. On the optimality of the FCC lattice for soft sphere packing. SIAM J. Discrete Math., 32(1):750–782, 2018.
L. Fejes Tóth. Some packing and covering theorems. Acta Sci. Math. (Szeged), 12:62–67, 1950.
T. C. Hales and S. McLaughlin. The Dodecahedral Conjecture. J. Amer. Math. Soc., 23:299–344, 2010.
T. Lambert. Empty shape triangulation algorithms. PhD thesis, University of Manitoba, 1994.
J. Molnár. On the 𝜌-system of unit circles. Ann. Univ. Sci. Budapest, Eötvös Sect. Math., 20:195–203, 1977.
C. A. Rogers. The closest packing of convex two-dimensional domains. Acta Math., 86:309–321, 1951.
C. A. Rogers. Erratum to: The closest packing of convex two-dimensional domains, corrigendum. Acta Math., 104:305–306, 1960.
Ch. Zong. The simultaneous packing and covering constants in the plane. Adv. Math., 218(3):653–672, 2008.