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

Abstract:

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