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

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 $\min\{f(\|x-y\|)\, :\, x,y\in \R^d\}$, where the minimum is taken over point sets of cardinality $N.$ We also show that $N^{1/d}Δ_d^{-1/d}-2\le D_d(N)\le N^{1/d}Δ_d^{-1/d}, \quad N\ge 2,$ where $Δ_d$ is the maximal sphere packing density in $\R^d$. Further, we provide asymptotic estimates for the $f$-best-packing constants as $N\to\infty$.

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