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Bounds for the minimum diameter of integral point sets
- Source :
- The Australasian Journal of Combinatorics, Vol. 39, Pages 233-240, 2007
- Publication Year :
- 2008
-
Abstract
- Geometrical objects with integral sides have attracted mathematicians for ages. For example, the problem to prove or to disprove the existence of a perfect box, that is, a rectangular parallelepiped with all edges, face diagonals and space diagonals of integer lengths, remains open. More generally an integral point set $\mathcal{P}$ is a set of $n$ points in the $m$-dimensional Euclidean space $\mathbb{E}^m$ with pairwise integral distances where the largest occurring distance is called its diameter. From the combinatorial point of view there is a natural interest in the determination of the smallest possible diameter $d(m,n)$ for given parameters $m$ and $n$. We give some new upper bounds for the minimum diameter $d(m,n)$ and some exact values.<br />Comment: 8 pages, 7 figures; typos corrected
- Subjects :
- Mathematics - Combinatorics
52C10, 11D99
Subjects
Details
- Database :
- arXiv
- Journal :
- The Australasian Journal of Combinatorics, Vol. 39, Pages 233-240, 2007
- Publication Type :
- Report
- Accession number :
- edsarx.0804.1296
- Document Type :
- Working Paper