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

TL;DR

This paper investigates the minimal N-point diameters and f-best-packing constants in Euclidean space, providing bounds, asymptotic estimates, and connecting these concepts to sphere packing densities.

Contribution

It introduces a unified framework for analyzing N-point diameters and f-best-packing constants, deriving bounds and asymptotics that link to sphere packing densities.

Findings

Derived bounds for N-point diameters in R^d.

Established asymptotic behavior of f-best-packing constants.

Connected packing constants to sphere packing density.

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 $\min\{f(\|x-y\|)\, :\, x,y\in \R^d\}$, 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 $N\to\infty$.