Covering and Packing with Spheres by Diagonal Distortion in ℝ n

On the Optimality of the FCC Lattice for Soft Sphere Packing

“…Subtracting (26) from (25), we get (23). Alternatively, we would start with (12) and get (23) directly. We prefer the latter, geometric argument to prove (24).…”

“…We consider the diagonal family of lattices introduced in [12]. Each lattice in this family is obtained by compressing or stretching the integer lattice, Z 3 , along the diagonal direction.…”

“…Departing from [2,18] and also [11], we introduce the soft density, δ 1 = ϕ 1 − i≥2 ϕ i , which penalizes for gaps in the coverage but also for overlaps among the balls. Following [12], we focus on the 1-parameter diagonal family of lattices obtained by compressing or stretching the integer lattice along the diagonal direction. We use the unimodality of the measure to prove the following optimality result in R 3 : THEOREM 1 (Main Result).…”

Delaunay simplices in diagonally distorted lattices

“…Lemma 2 in [EK11] shows that T 0 (Z d ) is isometric to T δ (Z d−1 ) for δ = 1/ √ d. Using Lemma 8 along with this fact, we arrive at the conclusion that…”

“…This family was first studied in [EK12] by Edelsbrunner and Kerber, with an ulterior motive to do topological analysis of high-dimensional image data. Later, this lattice was used to study covering and packing problems of Euclidean balls in different contexts [EIH18,EK11,IHKU14].…”