Minimal 𝑁-point diameters and 𝑓-best-packing constants in ℝ^{𝕕}

Abstract

In terms of the minimal N N -point diameter D d ( N ) D_d(N) for R d , \mathbb {R}^d, we determine, for a class of continuous real-valued functions f f on [ 0 , + ∞ ] , [0,+\infty ], the N N -point f f -best-packing constant min { f ( ‖ x − y ‖ ) : x , y ∈ R d } \min \{f(\|x-y\|)\, :\, x,y\in \mathbb {R}^d\} , where the minimum is taken over point sets of cardinality N . N. We also show that \[ N 1 / d Δ d − 1 / d − 2 ≤ D d ( N ) ≤ N 1 / d Δ d − 1 / d , N ≥ 2 , N^{1/d}\Delta _d^{-1/d}-2\le D_d(N)\le N^{1/d}\Delta _d^{-1/d}, \quad N\ge 2, \] where Δ d \Delta _d is the maximal sphere packing density in R d \mathbb {R}^d . Further, we provide asymptotic estimates for the f f -best-packing constants as N → ∞ N\to \infty .