# The maximum number of points in the cross-polytope that form a packing set of a scaled cross-polytope

## TL;DR

This paper investigates the maximum number of points that can be placed within a scaled cross-polytope such that the $l_1$-distance between any two points exceeds a certain threshold, providing exact bounds and constructions in various dimensions.

## Contribution

It determines the maximum packing sizes within scaled cross-polytopes for specific ranges of the scale parameter in multiple dimensions, including explicit constructions.

## Key findings

- Maximum of 2n points in n-dimensional cross-polytope for certain r
- Exact maximum points in 3D for specific r ranges
- Constructive arrangements achieving upper bounds

## Abstract

The problem of finding the largest number of points in the unit cross-polytope such that the $l_{1}$-distance between any two distinct points is at least $2r$ is investigated for $r\in\left(1-\frac{1}{n},1\right]$ in dimensions $\geq2$ and for $r\in\left(\frac{1}{2},1\right]$ in dimension $3$. For the $n$-dimensional cross-polytope, $2n$ points can be placed when $r\in\left(1-\frac{1}{n},1\right]$. For the three-dimensional cross-polytope, $10$ and $12$ points can be placed if and only if $r\in\left(\frac{3}{5},\frac{2}{3}\right]$ and $r\in\left(\frac{4}{7},\frac{3}{5}\right]$ respectively, and no more than $14$ points can be placed when $r\in\left(\frac{1}{2},\frac{4}{7}\right]$. Also, constructive arrangements of points that attain the upper bounds of $2n$, $10$, and $12$ are provided, as well as $13$ points for dimension $3$ when $r\in\left(\frac{1}{2},\frac{6}{11}\right]$.