ACTA issues

Maximum parametric soft density of lattice configurations of balls

Sami Mezal Almohammad

Acta Sci. Math. (Szeged) 87:3-4(2021), 615-647

Abstract. In 2018, Edelsbrunner and Iglesias-Ham defined a notion of density, called first soft density, for lattice packings of congruent balls in Euclidean $3$-space, which penalizes gaps and multiple overlaps. In their paper, they showed that this density is maximal in a $1$-parameter family of lattices, called diagonal family, for a configuration of congruent balls whose centers are the points of a face-centered cubic lattice. In this note we extend their notion of density, which we call first soft density of weight $t$, and show that it is maximal in the diagonal family for some family of congruent balls centered at the points of a face-centered cubic lattice, for every $t \geq 1$, and at the points of a body-centered cubic lattice for $t=0.5$.

DOI: 10.14232/actasm-020-483-y

AMS Subject Classification (1991): 52C17, 52A38, 52A15

Keyword(s): packing and covering, soft density of weight $t$, lattice configurations, Voronoi domains, Brillouin zones

received 2.12.2020, revised 23.4.2021, accepted 17.8.2021. (Registered under 233/2020.)