of our work. In a sequel to this paper, we will present and analyze DLP optimal packings and their corresponding R$_{min}$ ( N ) for the d = 3 case over a similar range of N . We will compare R$_{min}$ ( N ) values to shell distances in Barlow packings [20], where the Barlow packings, of which the best-known are face centered cubic (FCC) and hexagonal close packed (HCP) arrangements, all individually achieve the maximal infinite-volume packing fraction for d = 3, φ ∞ ∗ = π/ √ 18 = 0 . 740481 . . . . The coordinates for and images of DLP optimal packings and values for R$_{min}$ ( N ) over the entire range of N studied can be found on the authors’ website [21]. The realizability condition on g$_{2}$ ( r ) results from a relation between an upper bound on the maximal value of the function Z ( R ), to be defined shortly, and g$_{2}$ ( r ). The function Z ( r$_{i}$ , R ) is defined for packings of nonoverlapping spheres of unit diameter as the number of sphere centers that are within distance R from a (additional) sphere center at position r$_{i}$ , with i an index over centers. The maximum over all r$_{i}$ of Z ( r$_{i}$ , R ) is an upper bound on the maximum of the (different) function Z ( R ), where Z ( R ) is defined for a statistically homogeneous packing as the expected number of sphere centers within distance R from any given sphere center, or equivalently as the average of Z ( r$_{i}$ , R ) over all i . The function Z ( R ) can be related to the pair correlation function g$_{2}$ ( r ), where for a pair correlation function g$_{2}$ ( r ) that is direction-dependent, g$_{2}$ ( r ) is the directional average of g$_{2}$ ( r ), by In Eq. (3), ρ is the constant number density of sphere centers and s$_{1}$ ( r ) is the surface area of a sphere of radius r in$_{R}$ $^{d}$, The maximum at fixed R of the function Z ( r$_{i}$ , R ) over all possible configurations of sphere centers { r$_{i}$ } is equal to the greatest number N for which R$_{min}$ ( N ) ≤ R for a DLP optimal 6 s$_{1}$ ( r ) = 2 π d/ $^{2}$r d − 1 Γ( d/ 2) . (4) Z ( R ) = ρs$_{1}$ (1)$^{∫}$ R 0 x d − $^{1}$g$_{2}$ ( x ) dx. (3) II. PAIR CORRELATION FUNCTION REALIZABILITY AND Z$_{max}$ ( R )