[1908.05650] The Maximum Number Of Points In The Cross-polytope ...

Mathematics > Metric Geometry arXiv:1908.05650 (math) [Submitted on 15 Aug 2019] Title:The maximum number of points in the cross-polytope that form a packing set of a scaled cross-polytope Authors:Ji Hoon Chun View a PDF of the paper titled The maximum number of points in the cross-polytope that form a packing set of a scaled cross-polytope, by Ji Hoon Chun View PDF

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 $r\in\left(\frac{1}{2},\frac{6}{11}\right]$.

Subjects:	Metric Geometry (math.MG)
Cite as:	arXiv:1908.05650 [math.MG]
(or arXiv:1908.05650v1 [math.MG] for this version)
https://doi.org/10.48550/arXiv.1908.05650 Focus to learn more arXiv-issued DOI via DataCite

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From: Ji Hoon Chun [view email] [v1] Thu, 15 Aug 2019 17:47:21 UTC (1,747 KB) Full-text links:

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