  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 $$ N^{1/d}\Delta_d^{-1/d}-2\le D_d(N)\le N^{1/d}\Delta_d^{-1/d}, \quad
N\ge 2,$$ where $\Delta_d$ is the maximal sphere packing density in $\R^d$.
Further, we provide asymptotic estimates for the $f$-best-packing constants as
$N\to\infty$.
