We show that any total preorder on a set with $\binom{n}{2}$ elements coincides with the order on pairwise distances of some point collection of size $n$ in $\mathbb{R}^{n-1}$. For linear orders, a collection of $n$ points in $\mathbb{R}^{n-2}$ suffices. These bounds turn out to be optimal. We also find an optimal bound in a bipartite version for total preorders and a near-optimal bound for a bipartite version for linear orders. Our arguments include tools from convexity and positive semidefinite quadratic forms.

翻译：我们证明，对于任意具有$\binom{n}{2}$个元素的集合，其上的全预序均可与$\mathbb{R}^{n-1}$空间中某$n$个点构成点集的两两距离顺序相一致。对于线性序，$\mathbb{R}^{n-2}$空间中的$n$个点集即已足够。这些界被证明是最优的。我们进一步在全预序的二部版本中得到了最优界，并在线性序的二部版本中得到了接近最优的界。我们的论证运用了凸性理论与半正定二次型相关工具。