Minimal $N$-Point Diameters and $f$-Best-Packing Constants in $R^d$

Abstract

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