Bounds for the minimum diameter of integral point sets

Details Bounds for the minimum diameter of integral point sets Bounds for the minimum diameter of integral point sets

2008-04-08

arxiv.org/abs/0804.1296v1

Mathematics

Combinatorics

8 pages, 7 figures

Scientific paper

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.

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